On the thinness and proper thinness of a graph

TL;DR

This paper introduces the concept of proper thinness as a generalization of thinness in graphs, studies their properties, and explores algorithmic implications for problems on these graph classes.

Contribution

It defines proper thinness, analyzes its properties, and extends polynomial-time solvability results to broader classes of problems on these graphs.

Findings

Proper thinness generalizes proper interval graphs.

Certain problems are solvable in polynomial time on graphs with bounded thinness.

Extended polynomial-time algorithms for problems on graphs with bounded proper thinness.

Abstract

Graphs with bounded thinness were defined in 2007 as a generalization of interval graphs. In this paper we introduce the concept of proper thinness, such that graphs with bounded proper thinness generalize proper interval graphs. We study the complexity of problems related to the computation of these parameters, describe the behavior of the thinness and proper thinness under three graph operations, and relate thinness and proper thinness with other graph invariants in the literature. Finally, we describe a wide family of problems that can be solved in polynomial time for graphs with bounded thinness, generalizing for example list matrix partition problems with bounded size matrix, and enlarge this family of problems for graphs with bounded proper thinness, including domination problems.

Full text

1 Introduction

A graph $G=(V,E)$ is $k$-thin if there exist an ordering $v_{1},\dots,v_{n}$ of $V$ and a partition of $V$ into $k$ classes $(V^{1},\dots,V^{k})$ such that, for each triple $(r,s,t)$ with $r<s<t$, if $v_{r}$, $v_{s}$ belong to the same class and $v_{t}v_{r}\in E$, then $v_{t}v_{s}\in E$. The minimum $k$ such that $G$ is $k$-thin is called the thinness of $G$. The thinness is unbounded on the class of all graphs, and graphs with bounded thinness were introduced in [29] as a generalization of interval graphs, which are exactly the $1$-thin graphs. When a representation of the graph as a $k$-thin graph is given, for a constant value $k$, some NP-complete problems as maximum weighted independent set and bounded coloring with fixed number of colors can be solved in polynomial time [29, 5]. These algorithms were respectively applied for improving heuristics of two real-world problems: the Frequency Assignment Problem in GSM networks [29], and the Double Traveling Salesman Problem with Multiple Stacks [5]. In this work we propose a framework to describe a wide family of problems that can be solved by dynamic programming techniques on graphs with bounded thinness, when the $k$-thin representation of the graph is given. These problems generalize for example the list matrix partition problems for matrices of bounded size [14].

We introduce here the concept of proper thinness of a graph with the aim of generalizing proper-interval graphs: graphs that are proper $1$-thin are exactly proper interval graphs (see Section 2 for a definition). We extend the framework in order to solve in polynomial time by dynamic programming many of the domination-type problems in the literature (e.g. classified in [1]) and their weighted versions, such as existence/minimum (weighted) independent dominating set, minimum (weighted) total dominating set, minimum perfect dominating set and existence/minimum (weighted) efficient dominating set, for the class of graphs with bounded proper thinness $k$, when the proper $k$-thin representation of the graph is given.

The organization of the paper is the following. In Section 2 we state the main definitions and present some basic results on thinness. In Section 3, we study some problems related to the recognition of $k$-thin graphs and proper $k$-thin graphs. We analyze the computational complexity of finding a suitable vertex partition when a vertex ordering is given, and, conversely, finding a vertex ordering when a vertex partition is given.

In Section 4 we survey the relation of thinness and other width parameters in graphs. In Section 5 we relate the proper thinness of interval graphs to other interval graph invariants, as interval count and chains of nested intervals.

In Section 6 we describe a wide family of problems that can be solved in polynomial time for graphs with bounded thinness, when the representation is given. In Section 6.1 we extend that family to include dominating-like problems that can be solved in polynomial time for graphs with bounded proper thinness.

In Section 7 we describe the behavior of the thinness and proper thinness under three graph operations: union, join, and Cartesian product. The first two results allow us to fully characterize $k$-thin graphs by forbidden induced subgraphs within the class of cographs. The third result is used to show the polynomiality of the $t$-rainbow domination problem for fixed $t$ on graphs with bounded thinness.

2 Definitions and basic results

All graphs in this work are finite, undirected, and have no loops or multiple edges. For all graph-theoretic notions and notation not defined here, we refer to West [45]. Let $G$ be a graph. Denote by $V(G)$ its vertex set, by $E(G)$ its edge set, by $\overline{G}$ its complement, by $N(v)$ the neighborhood of a vertex $v$ in $G$, by $N[v]$ the closed neighborhood $N(v)\cup\{v\}$, and by $\overline{N}(v)$ the non-neighbors of $v$. If $X\subseteq V(G)$, denote by $N(X)$ the set of vertices not in $X$ having at least one neighbor in $X$.

Denote by $G[W]$ the subgraph of $G$ induced by $W\subseteq V(G)$, and by $G-W$ or $G\setminus W$ the graph $G[V(G)\setminus W]$. A subgraph $H$ (not necessarily induced) of $G$ is a spanning subgraph if $V(H)=V(G)$.

Denote the size of a set $S$ by $|S|$. A clique (resp. stable set) is a set of pairwise adjacent (resp. nonadjacent) vertices. We use maximum to mean maximum-sized, whereas maximal means inclusion-wise maximal. The use of minimum and minimal is analogous.

Denote by $K_{n}$ the graph induced by a clique of size $n$. A claw is the graph isomorphic to $K_{1,3}$. Let $H$ be a graph and $t$ a natural number. The disjoint union of $t$ copies of the graph $H$ is denoted by $tH$.

For a positive integer $r$, the $(r\times r)$-grid is the graph whose vertex set is $\{(i,j):1\leq i,j\leq r\}$ and whose edge set is $\{(i,j)(k,l):|i-k|+|j-l|=1,\mbox{ where }1\leq i,j,k,l\leq r\}$.

A dominating set in a graph is a set of vertices such that each vertex outside the set has at least one neighbor in the set.

A coloring of a graph is an assignment of colors to its vertices such that any two adjacent vertices are assigned different colors. The smallest number $t$ such that $G$ admits a coloring with $t$ colors (a $t$-coloring) is called the chromatic number of $G$ and is denoted by $\chi(G)$. A coloring defines a partition of the vertices of the graph into stable sets, called color classes. List variations of the vertex coloring problem can be found in the literature. For a survey on that kind of related problems, see [42]. In the list-coloring problem, every vertex $v$ comes equipped with a list of permitted colors $L(v)$ for it.

For a symmetric matrix $M$ over $0,1,*$, the $M$-partition problem seeks a partition of the vertices of the input graph into independent sets, cliques, or arbitrary sets, with certain pairs of sets being required to have no edges, or to have all edges joining them, as encoded in the matrix $M$: $M_{ii}=1$ means the $i$-th set is a clique, while $M_{ii}=0$ means the $i$-th set is a stable set; for $i\neq j$, $M_{ij}=1$ means every vertex of the $i$-th set is adjacent to every vertex of the $j$-th set, while $M_{ij}=0$ means there are no edges from the $i$-th set to the $j$-th set. Moreover, the vertices of the input graph can be equipped with lists, restricting the parts to which a vertex can be placed. In that case the problem is know as a list matrix partition problem. Such (list) matrix partition problems generalize (list) colorings and (list) homomorphisms [14].

When discussing about algorithms and data structures, we denote by $n$ the number of vertices of the input graph $G$.

Given a graph $G$, a weight function $w$ on $V(G)$, and a subset $S\subseteq V(G)$, the weight of $S$, denoted by $w(S)$ is defined as $\sum_{v\in S}w(v)$.

A class of graphs is hereditary when if a graph $G$ is in the class, then every induced subgraph of $G$ is in the same class.

A graph is a cograph if it contains no induced path of length four.

A graph $G(V,E)$ is a comparability graph if there exists an ordering $v_{1},\dots,v_{n}$ of $V$ such that, for each triple $(r,s,t)$ with $r<s<t$, if $v_{r}v_{s}$ and $v_{s}v_{t}$ are edges of $G$, then so is $v_{r}v_{t}$. Such an ordering is a comparability ordering. A graph is a co-comparability graph if its complement is a comparability graph.

A graph $G$ is an interval graph if to each vertex $v\in V(G)$, can be associated a closed interval $I_{v}=[l_{v},r_{v}]$ of the real line, such that two distinct vertices $u,v\in V(G)$ are adjacent if and only if $I_{u}\cap I_{v}\neq\emptyset$. The family $\{I_{v}\}_{v\in V(G)}$ is an interval representation of $G$. An undirected graph $G$ is a proper interval graph if there is an interval representation of $G$ in which no interval properly contains another. In the same way, an undirected graph $G$ is a unit interval graph if there is an interval representation of $G$ in which all the intervals have the same length.

In 1969, Roberts [35] proved that the classes of proper interval graphs, unit interval graphs, and interval graphs with no claw as induced subgraph coincide.

The right-end ordering of the vertices of an interval graph satisfies the following property: for each triple $(r,s,t)$ with $r<s<t$, if $v_{t}v_{r}\in E$, then $v_{t}v_{s}\in E$. In other words, the neighbors of vertex $t$ with index less than $t$ are $t-1,t-2,\dots,t-d$ for some $d\geq 0$. Moreover, a graph $G$ is an interval graph if and only if there exists an ordering of its vertices satisfying the property above [34, 32].

Let us repeat and extend the definition of $k$-thinness given in the introduction. A graph $G=(V,E)$ is $k$-thin if there exist an ordering $v_{1},\dots,v_{n}$ of $V$ and a partition of $V$ into $k$ classes such that, for each triple $(r,s,t)$ with $r<s<t$, if $v_{r}$, $v_{s}$ belong to the same class and $v_{t}v_{r}\in E$, then $v_{t}v_{s}\in E$. An ordering and a partition satisfying those properties are said to be consistent. The minimum $k$ such that $G$ is $k$-thin is called the thinness of $G$ and denoted by $\operatorname{thin}(G)$.

The thinness of a graph was introduced by Mannino, Oriolo, Ricci, and Chandran in 2007 [29]. Graphs with bounded thinness (thinness bounded by a constant value) are a generalization of interval graphs, that are exactly the graphs with thinness 1, and capture some of their algorithmic properties.

Let $\overline{tK_{2}}$ be the complement of a matching of size $t$.

Theorem 1

[29]* For every $t\geq 1$, $\operatorname{thin}(\overline{tK_{2}})=t$.*

The right-end ordering of the vertices of a proper interval graph satisfies the following property: for each triple $(r,s,t)$ with $r<s<t$, if $v_{t}v_{r}\in E$, then $v_{t}v_{s}\in E$ and $v_{r}v_{s}\in E$. In other words, the neighbors of vertex $t$ with index less than $t$ are $t-1,t-2,\dots,t-d$, and those with index greater than $t$ are $t+1,t+2,\dots,t+d^{\prime}$. Moreover, $G$ is a proper interval graph if and only if there exists an ordering of its vertices satisfying the property above [12, 28].

We define the concept of proper thinness of graphs as follows.

A graph $G=(V,E)$ is proper $k$-thin if there exist an ordering $v_{1},\dots,v_{n}$ of $V$ and a partition of $V$ into $k$ classes $(V^{1},\dots,V^{k})$ such that, for each triple $(r,s,t)$ with $r<s<t$, if $v_{r}$, $v_{s}$ belong to the same class and $v_{t}v_{r}\in E$, then $v_{t}v_{s}\in E$ and if $v_{s}$, $v_{t}$ belong to the same class and $v_{r}v_{t}\in E$, then $v_{r}v_{s}\in E$. Equivalently, $G$ is proper $k$-thin if both $v_{1},\dots,v_{n}$ and $v_{n},\dots,v_{1}$ are consistent with the partition. In this case, the partition and the ordering $v_{1},\dots,v_{n}$ are said to be strongly consistent, and the minimum $k$ such that $G$ is proper $k$-thin is called the proper thinness of $G$ and denoted by $\operatorname{pthin}(G)$.

Since $k$-thin graphs are defined as a generalization of interval graphs, proper $k$-thin graphs arise naturally as a generalization of proper interval graphs. It can be seen that a graph is proper $1$-thin if and only if it is a proper interval graph. Moreover, the proper-thinness of the class of interval graphs is unbounded (See Proposition 10).

3 Algorithmic aspects

We will deal in this section with some questions related to the recognition problem of (proper) $k$-thin graphs. The recognition problem itself is open so far for both classes, but we will show that, given a vertex ordering of a graph, we can find in polynomial time a partition into a minimum number of classes which is (strongly) consistent with the ordering. On the other hand, we will show that given a graph and a vertex partition, it is NP-complete to decide if there exists an ordering of the vertices of the graph which is (strongly) consistent with the partition.

Theorem 2

Given a graph $G$ and an ordering ${<}$ of its vertices, one can find in polynomial time graphs $G_{<}$ and $\tilde{G}_{<}$ with the following properties:

1. (1)

$V(G_{<})=V(\tilde{G}_{<})=V(G)$; 2. (2)

the chromatic number of $G_{<}$ (resp. $\tilde{G}_{<}$) is equal to the minimum integer $k$ such that there is a partition of $V(G)$ into $k$ sets that is consistent (resp. strongly consistent) with the order $<$, and the color classes of a valid coloring of $G_{<}$ (resp. $\tilde{G}_{<}$) form a partition consistent (resp. strongly consistent) with ${<}$; 3. (3)

$G_{<}$* and $\tilde{G}_{<}$ are co-comparability graphs.*

In particular, the minimum integer $k$ as in (2) and a partition into $k$ vertex sets can be computed in polynomial time. Moreover, if $G$ is a co-comparability graph and ${<}$ a comparability ordering of $\overline{G}$, then $G_{<}$ and $\tilde{G}_{<}$ are spanning subgraphs of $G$.

Proof. Let $G$ be a graph and ${<}$ an ordering of its vertices. We will build a graph $G_{<}$ such that $V(G_{<})=V(G)$, and $v<w$ are adjacent in $G_{<}$ if and only if they cannot belong to the same class of a partition which is consistent with $<$. By definition of consistency, this happens if and only if there is a vertex $z$ in $G$ such that $v<w<z$, $z$ is adjacent to $v$ and nonadjacent to $w$. So define $E(G_{<})$ such that for $v<w$, $vw\in E(G_{<})$ if and only if there is a vertex $z$ in $G$ such that $v<w<z$, $zv\in E(G)$ and $zw\not\in E(G)$.

We build $\tilde{G}_{<}$ in a similar way. In this case, for $v<w$, $vw\in E(\tilde{G}_{<})$ if and only if either there is a vertex $z$ in $G$ such that $v<w<z$, $zv\in E(G)$ and $zw\not\in E(G)$ or there is a vertex $x$ in $G$ such that $x<v<w$, $xw\in E(G)$ and $xv\not\in E(G)$.

Let us see that $<$ is a comparability ordering both for $\overline{G_{<}}$ and $\overline{\tilde{G}_{<}}$. Suppose on the contrary that there is a triple $r<s<t$ in $V(G)$ such that $rs$, $st$ are edges of $\overline{G_{<}}$ (resp. $\overline{\tilde{G}_{<}}$) and $rt$ is not an edge of $\overline{G_{<}}$ (resp. $\overline{\tilde{G}_{<}}$). By definition of $G_{<}$ (resp. $\tilde{G}_{<}$), there is a vertex $z$ such that $r<s<t<z$, $zr\in E(G)$ and $zt\not\in E(G)$ (resp. either there is a vertex $z$ such that $r<s<t<z$, $zr\in E(G)$ and $zt\not\in E(G)$, or there is a vertex $x$ in $G$ such that $x<r<s<t$, $xt\in E(G)$ and $xr\not\in E(G)$). If $zs\not\in E(G)$, then $rs$ is an edge of $G_{<}$ (resp. $\tilde{G}_{<}$), a contradiction. If $zs\in E(G)$, then $st$ is an edge of $G_{<}$ (resp. $\tilde{G}_{<}$), a contradiction as well. The case of $x$ for $\tilde{G}_{<}$ is symmetric, if $xs\not\in E(G)$, then $st$ is an edge of $\tilde{G}_{<}$, a contradiction. If $xs\in E(G)$, then $rs$ is an edge of $\tilde{G}_{<}$, a contradiction as well. So $G_{<}$ and $\tilde{G}_{<}$ are co-comparability graphs, being $<$ a comparability ordering for $\overline{G_{<}}$ and $\overline{\tilde{G}_{<}}$, respectively.

As we have defined $G_{<}$ (resp. $\tilde{G}_{<}$) such that $V(G_{<})=V(\tilde{G}_{<})=V(G)$, and $v<w$ are adjacent in $G_{<}$ (resp. $\tilde{G}_{<}$) if and only if they cannot belong to the same class of a partition which is consistent (resp. strongly consistent) with $<$, it follows that there is a one-to-one correspondence between partitions of $V(G)$ consistent (resp. strongly consistent) with $<$ and colorings of $G_{<}$ (resp. $\tilde{G}_{<}$). In particular, the minimum $k$ such that there is a partition of $V(G)$ into $k$ sets that is consistent (resp. strongly consistent) with $<$ is the chromatic number of $G_{<}$ (resp. $\tilde{G}_{<}$). An optimum coloring of $G_{<}$ (resp. $\tilde{G}_{<}$) can be computed in polynomial time [16].

To complete the proof of the theorem, suppose now that $G$ is a co-comparability graph and $<$ is a comparability ordering for $\overline{G}$. Let $v<w$ adjacent in $G_{<}$ (resp. $\tilde{G}_{<}$). By definition, there is a vertex $z$ in $G$ such that $v<w<z$, $vz\in E(G)$ and $wz\not\in E(G)$ (resp. either there is a vertex $z$ in $G$ such that $v<w<z$, $vz\in E(G)$ and $wz\not\in E(G)$, or there is a vertex $x$ in $G$ such that $x<v<w$, $xw\in E(G)$ and $xv\not\in E(G)$). If $vw\not\in E(G)$, being $\overline{G}$ a comparability graph, $vz\not\in E(G)$, a contradiction. So $vw\in E(G)$. This proves that $G_{<}$ is a spanning subgraph of $G$. The case of $x$ for $\tilde{G}_{<}$ is symmetric, if $vw\not\in E(G)$, being $\overline{G}$ a comparability graph, $xw\not\in E(G)$, a contradiction. So in any case $vw\in E(G)$. This proves that $\tilde{G}_{<}$ is a spanning subgraph of $G$ as well. $\Box$

A direct consequence of this result is the following, that was already proved in [5] for the case of thinness.

Corollary 3

If $G$ is a co-comparability graph, $\operatorname{thin}(G)\leq\operatorname{pthin}(G)\leq\chi(G)$. Moreover, any vertex partition given by a coloring of $G$ and any comparability ordering for its complement are strongly consistent.

As already observed in [5], the bound $\operatorname{thin}(G)\leq\operatorname{pthin}(G)\leq\chi(G)$ for co-comparability graphs can be arbitrarily bad: for example, if $G$ is a clique of size $n$, then $\operatorname{thin}(G)=\operatorname{pthin}(G)=1$ and $\chi(G)=n$. However, it holds with equality for graphs $\overline{tK_{2}}$, because $\operatorname{thin}(\overline{tK_{2}})=\operatorname{pthin}(\overline{tK_{2}})=\chi(\overline{tK_{2}})=t$ (Theorem 1 and Corollary 3).

In contrast with Theorem 2, if a partition is given, it is NP-complete to decide the existence of a (strongly) consistent ordering.

(Strongly) Consistent ordering with a given partition

Instance: A graph $G=(V,E)$ and a partition of $V$ into non-empty subsets.

Question: Does there exist a total order $<$ of $V$ (strongly) consistent with the given partition?

The proof is based on a reduction from the following problem, which is known to be NP-complete [18].

Non-Betweenness

Instance: A finite set $A$ and a collection $S$ of ordered triples of distinct elements of $A$.

Question: Does there exist a total order $<$ of $A$ such that for each $(x,y,z)\in S$, it is never the case that $x<y<z$ or $z<y<x$ (i.e. $y$ is not between $x$ and $z$)?

We start with an easy lemma.

Lemma 4

Let $G$ be a graph, $<$ an ordering of $V(G)$ and $V_{1},\dots,V_{k}$ a partition of $V(G)$ that is consistent with $<$. Let $\{x_{i},y_{i}\}\subseteq V_{i}$, for $i=1,2$, such that $x_{1}x_{2}$ and $y_{1}y_{2}$ are the only edges between $\{x_{1},y_{1}\}$ and $\{x_{2},y_{2}\}$. Then $x_{1}<y_{1}$ if and only if $x_{2}<y_{2}$.

Proof. By symmetry, let us assume that $y_{1}$ is the biggest vertex according to $<$. Again by symmetry, to prove the lemma it is enough to prove that $x_{2}<y_{2}$. By definition of consistency, since $x_{2}$ and $y_{2}$ are in the same class and $y_{1}$ is adjacent to $y_{2}$ but not to $x_{2}$, it is not possible that $y_{2}<x_{2}<y_{1}$. $\Box$

Theorem 5

The problem (Strongly) Consistent ordering with a given partition is NP-complete.

Proof. First note that (Strongly) Consistent ordering with a given partition is in NP, by using the total order of $V$ as the certificate.

Now let us prove its NP-hardness. Given an instance $(A,S)$ of Non-Betweenness, build a graph $G=(V,E)$ and a partition $V_{0},V_{1},\dots V_{|S|}$ of $V$ as follows.

Fix an ordering of the triples in $S$. Vertices of $V_{0}$ are in one-to-one correspondence with elements of $A$. For $i=1,\dots,|S|$, $V_{i}$ has $3$ vertices, and they are in a one-to-one correspondence with the elements of the $i$-th triple in $S$. Let us call $a^{i}$ the element of $V_{i}$ that corresponds to $a\in A$, for $i=0,\dots,|S|$.

Define the edges of $G$ as follows: for each triple $(x,y,z)\in S$, let $V_{i}$ be its corresponding set. The only edge in the subgraph induced by $\{x^{i},y^{i},z^{i}\}$ is $x^{i}z^{i}$. The remaining edges of $G$ are all the possible edges between vertices associated to the same $a\in A$.

Suppose first there is an ordering $<$ consistent with the partition $\{V_{0},$ $\dots,$ $V_{|S|}\}$. By Lemma 4, for each $1\leq i\leq|S|$, the relative order of the vertices $x^{i},y^{i},z^{i}$ is the same as the relative order of the vertices $x^{0},y^{0},z^{0}$. By definition of consistency and since the only edge in the subgraph induced by $\{x^{i},y^{i},z^{i}\}$ is $x^{i}z^{i}$, $y^{i}$ is not between $x^{i}$ and $z^{i}$ in that order. So the order of the vertices in $V_{0}$ gives a positive answer to the instance $(A,S)$ of Non-Betweenness.

Suppose now that there is a valid order $<$ for the instance $(A,S)$ of Non-Betweenness. We can extend $<$ to $V(G)$ by making consecutive all the copies in $V(G)$ of an element of $A$. Now, let $p<q<r$ be three vertices of $G$ such that $p$, $q$ belong to the same class $V_{i}$ and $rp\in E(G)$. Since $V_{0}$ is a stable set and the triples in $S$ satisfy the non-betweenness condition, $r$ is not in $V_{i}$. So $r$ and $p$ correspond to the same element $a$ of $A$, and since there is at most one copy of an element of $A$ in each $V_{i}$, $q$ does not correspond to a copy of $a$. But this contradicts the fact that all the vertices of $G$ that correspond to a same element of $A$ are consecutive. So the situation cannot arise, and the extended order is consistent with the partition. The case in which $q$, $r$ belong to the same class $V_{i}$ is identical, and indeed the extended order is strongly consistent with the partition. $\Box$

The computational complexity of the decision of existence of a (strongly) consistent ordering when the number of sets in the partition is fixed is still open. So is the computational complexity of deciding if a graph is (proper) $k$-thin, even for fixed $k\geq 2$. In the case of proper thinness, the problem is open even within the class of interval graphs.

4 Thinness and other width parameters

Many width parameters are defined in the literature. In this section we compile the results relating the thinness with some of them, namely pathwidth [37], treewidth [2, 38], clique-width [10], cutwidth [26], mim-width [43], and boxicity [36].

In [29] it was proved that the thinness of a graph is at most the pathwidth plus one, and that the gap may be high, since the pathwidth of a complete graph with $r$ vertices is $r-1$, while its thinness is $1$.

On the other hand, in [8] it was proved that the boxicity is a lower bound for the thinness of a graph, and it was pointed out that the difference can be large, as an $(r\times r)$-grid has boxicity $2$ and thinness $\Theta(r)$.

The vertex isoperimetric peak of a graph $G$, denoted as $b_{v}(G)$, is defined as $\max_{s}\min_{X\subset V,|X|=s}|N(X)|$. The thinness of the grid was estimated by using the following result, that was also used in [3] to give a lower bound of the thinness of a complete binary tree. We will use it as well to estimate the thinness of complete $m$-ary trees.

Lemma 6

[8]* For every graph $G$, $\operatorname{thin}(G)\geq b_{v}(G)/\Delta(G)$.*

Interval graphs have thinness $1$ and unbounded clique-width [17], while cographs have clique-width $2$ [11] and unbounded thinness, because $\overline{tK_{2}}$ is a cograph for every $t$, so the parameters are not comparable.

Complete graphs have high treewidth and thinness $1$, and trees instead have treewidth $1$ but we have the following result.

Theorem 7

For every fixed value $m$, the thinness of the complete $m$-ary tree on $n$ vertices is $\Theta(\log n)$.

Proof. In [44] it was proved that the vertex isoperimetric peak of the complete $m$-ary tree of height $h$ is $\Theta(h)$. On the other hand, it was proved in [13, 40] that the pathwidth of the complete $m$-ary tree of height $h$ is $\Theta(h)$. As the thinness of a graph is upper bounded by the pathwidth plus one [29] and using Lemma 6, it follows that the thinness of the complete $m$-ary tree of height $h$ is $\Theta(h)$, and this proves the theorem. $\Box$

The cutwidth of a graph $G$, denoted as $\operatorname{cutw}(G)$, is the smallest integer $k$ such that the vertices of $G$ can be arranged in a linear layout $v_{1},\dots,v_{n}$ in such a way that for every $i=1,\dots,n-1$, there are at most $k$ edges with one endpoint in $\{v_{1},\dots,v_{i}\}$ and the other in $\{v_{i+1},\ldots,v_{n}\}$.

Theorem 8

For every graph $G$, $\operatorname{thin}(G)\leq\operatorname{cutw}(G)+1$. Moreover, a linear layout realizing the cutwidth leads to a consistent partition into at most $\operatorname{cutw}(G)+1$ classes.

Proof. Let $G$ be a graph of cutwidth $k$, and let $v_{1},\dots,v_{n}$ such that for every $i=1,\dots,n-1$, there are at most $k$ edges with one endpoint in $\{v_{1},\dots,v_{i}\}$ and the other in $\{v_{i+1},\ldots,v_{n}\}$. Let $G_{<}$ be the graph defined as in Theorem 2 for the order $v_{1},\dots,v_{n}$. Since $G_{<}$ is a co-comparability graph, its chromatic number equals the size of a maximum clique of it [30]. Suppose that $G_{<}$ has a clique $H$ of size $k+2$, and let $v_{i}$ be the vertex of higher index in $H$. By definition of $G_{<}$, for each $i^{\prime}<i$ such that $v_{i^{\prime}}\in H$, there exists $j>i$ such that $v_{j}$ is adjacent to $v_{i^{\prime}}$ and not adjacent to $v_{i}$. So, there are at least $k+1$ edges with one endpoint in $\{v_{1},\dots,v_{i}\}$ and the other in $\{v_{i+1},\ldots,v_{n}\}$, a contradiction. $\Box$

The gap may be high, as for example on cliques.

The linear MIM-width of a graph $G$, denoted as $\operatorname{lmimw}(G)$, is the smallest integer $k$ such that the vertices of $G$ can be arranged in a linear layout $v_{1},\dots,v_{n}$ in such a way that for every $i=1,\dots,n-1$, the size of a maximum induced matching in the bipartite graph formed by the edges of $G$ with an endpoint in $\{v_{1},\dots,v_{i}\}$ and the other one in $\{v_{i+1},\ldots,v_{n}\}$ is at most $k$. This is the linear version of a parameter called MIM-width [43], that is a lower bound for the linear MIM-width.

Theorem 9

For every graph $G$, $\operatorname{lmimw}(G)\leq\operatorname{thin}(G)$. Moreover, a linear ordering $v_{1},\dots,v_{n}$ realizing the thinness, satisfies that the size of a maximum induced matching in the bipartite graph formed by the edges of $G$ with an endpoint in $\{v_{1},\dots,v_{i}\}$ and the other one in $\{v_{i+1},\ldots,v_{n}\}$ is at most $\operatorname{thin}(G)$.

Proof. Let $k=\operatorname{thin}(G)$ and consider a $k$-thin representation of $G$, with ordering $<$ of $V(G)$, namely $v_{1}<\dots<v_{n}$, and a partition of $V(G)$ into $k$ classes. Let $1\leq i\leq n-1$ and let $M$ be a maximum induced matching in the bipartite graph formed by the edges of $G$ with an endpoint in $\{v_{1},\dots,v_{i}\}$ and the other one in $\{v_{i+1},\ldots,v_{n}\}$. Suppose $v_{r}v_{t}$ and $v_{s}v_{q}$ belong to $M$, with $r<s\leq i$, $t,q\geq i+1$. If $v_{r}$ and $v_{s}$ belong to the same class of the partition, by definition of $k$-thin representation, $v_{s}v_{t}$ is also an edge, a contradiction with the fact that $M$ is an induced matching. So, $|M|\leq k$, thus $\operatorname{lmimw}(G)\leq\operatorname{thin}(G)$. $\Box$

As a corollary, given a graph $G$ provided with a $k$-thin representation, a wide family of problems known as Locally Checkable Vertex Subset and Vertex Partitioning Problems (LC-VSVP Problems) can be solved in $n^{(O(k))}$ time [43], as this holds for MIM-width $k$ and a suitable ordering. This family of problems is not comparable (inclusion-wise) with the one in Section 6, but encompasses maximum weighted independent set and minimum weighted dominating set.

5 Interval graphs with high proper thinness

In this section we first show that proper thinness of the class of interval graphs is unbounded. Then we relate the proper thinness of interval graphs to other interval graphs invariants, like interval count. A family of interval graphs with arbitrarily large proper thinness is the following.

Let $h\geq 1$, and define clawh as the graph obtained from the complete ternary tree of height $h$ by adding all the edges between a vertex of the tree and its ancestors. It is easy to see that $\mbox{claw}_{h}$ is an interval graph for every $h\geq 1$ (an interval representation of claw3 can be seen in Figure 1). The graph claw1 is the claw.

Proposition 10

[39]* For any $h\geq 1$, $\operatorname{pthin}(\mbox{claw}_{h})=h+1$.*

Proof. Let $h\geq 1$. We will label the vertices of $G=\mbox{claw}_{h}$ as $v^{i}_{j}$ such that $0\leq i\leq h$, $1\leq j\leq 3^{i}$, $v^{0}_{1}$ is the root of the ternary tree, and for each $0\leq i\leq h-1$, $1\leq j\leq 3^{i}$, the children of $v^{i}_{j}$ are $v^{i+1}_{3j-2}$, $v^{i+1}_{3j-1}$, and $v^{i+1}_{3j}$. Let us consider an ordering $<$ and a partition of $V(G)$ that are strongly consistent. Without loss of generality, by symmetry, we may assume $v^{i}_{2}<v^{i}_{1}<v^{i}_{3}$ for every $i\geq 1$.

Let us show now that for every $0\leq i^{\prime}<i\leq h$, $v^{i}_{1}$ and $v^{i^{\prime}}_{1}$ cannot be in the same class of the partition. Otherwise, if $v^{i}_{1}<v^{i^{\prime}}_{1}$ then the fact of $v^{i}_{2}<v^{i}_{1}<v^{i^{\prime}}_{1}$, $v^{i}_{2}v^{i^{\prime}}_{1}\in E(G)$ and $v^{i}_{2}v^{i}_{1}\not\in V(G)$ contradicts the definition of strong consistency, and if $v^{i^{\prime}}_{1}<v^{i}_{1}$ then the fact of $v^{i^{\prime}}_{1}<v^{i}_{1}<v^{i}_{3}$, $v^{i}_{3}v^{i^{\prime}}_{1}\in E(G)$ and $v^{i}_{3}v^{i}_{1}\not\in V(G)$ contradicts the definition of strong consistency.

So, $v^{0}_{1},\dots,v^{h}_{1}$ are all in different classes of the partition and $\operatorname{pthin}(\mbox{claw}_{h})\geq h+1$. On the other hand, a partition of the vertices according to its height in the tree, and a postorder of the vertices of the tree are strongly consistent. Thus $\operatorname{pthin}(\mbox{claw}_{h})=h+1$. $\Box$

This example is also a classical example of a graph with high interval count and high length of a chain of nested intervals. We will relate the proper thinness of interval graphs to these two interval graphs invariants.

The interval count of an interval graph $G$ is the minimum number of different interval sizes needed in an interval representation of $G$ (see for example [7, 25]). Graphs with interval count at most $k$ are also known as $k$-length interval graphs.

A $k$-nested interval graph is an interval graph admitting an interval representation in which there are no chains of $k+1$ intervals nested in each other [24]. It is easy to see that $k$-nested interval graphs are a superclass of $k$-length interval graphs. We have also the following property.

Proposition 11

[31]* Every $k$-nested interval graph is proper $k$-thin.*

Proof. Let $G$ be a $k$-nested interval graph and consider an interval representation of $G$ with no chains of $k+1$ intervals nested in each other. It is a known result that we may assume that all the interval endpoints are distinct. We label each interval by the length of the longest chain of nested intervals ending in it, and these labels define the partition of the vertices into classes, that are at most $k$. Now, we order the vertices according to their intervals by the right endpoint (left to right). That order is consistent with the partition in which the only class contains all vertices of $G$, so, in particular, it is consistent with every other partition refining it. Let us see that the consistency is strong. Let $r<s<t$ such that $s$ and $t$ are in the same class of the partition. Let $I_{r},I_{s},I_{t}$ their corresponding intervals. By definition of the classes, $I_{s}\not\subseteq I_{t}$, otherwise the length of the longest chain of nested intervals ending in $I_{s}$ would be strictly greater than the one for $I_{t}$. As the right endpoint of $I_{t}$ is greater than the one of $I_{s}$, it follows that the left endpoint of $I_{t}$ is also greater than the one of $I_{s}$. Thus, if $I_{r}$ intersects $I_{t}$, it intersects $I_{s}$ as well. So, the ordering and the partition are strongly consistent and $G$ is proper $k$-thin. $\Box$

Graphs with interval count one are known as unit interval graphs, while $1$-nested interval graph are equivalent to proper interval graphs. In [35] it is shown that unit interval graphs are equivalent to proper interval graphs. So the classes proper $1$-thin, $1$-length interval and $1$-nested interval are equivalent. We will see that for higher numbers the equivalence does not necessarily hold.

Indeed, in [15, Theorem 5, p. 177], Fishburn shows that, for every $k\geq 2$, there are $2$-nested interval graphs that are not $k$-length interval.

We will describe a family of graphs that show that, for every $k\geq 3$, there are proper $3$-thin graphs that are not $k$-nested interval.

Let $k\geq 1$. Let $G_{k}$ with $3k+1$ vertices is defined as follows. Its vertex-set is $V_{k}=A_{k}\cup B_{k}\cup W_{k}$, where $A_{k}=\{a_{1},\dots,a_{k}\}$, $B_{k}=\{b_{1},\dots,b_{k}\}$ and $W_{k}=\{v_{1},\dots,v_{k},v_{k+1}\}$. The subgraph induced by $W_{k}$ is a clique with $k+1$ vertices; $a_{1}$ (resp., $b_{1}$) is adjacent to $v_{1}$. Then, for any $1<i\leq k$, $a_{i}$ (resp., $b_{i}$) is adjacent to $a_{i-1}$ (resp., to $b_{i-1}$), and to $v_{j}$ for any $j\geq i$. See Figure 2 for a sketch of $G_{k}$ and an interval representation of it.

The graph $G_{1}$ is the claw, which is not proper interval. For higher values of $k$, we have the following property.

Proposition 12

[31]* For any $k\leq 2$, $G_{k}$ is proper $3$-thin, but in every interval representation of it, if $I_{j}$ is the interval corresponding to $v_{j}$, it holds $I_{k+1}\subseteq I_{k}\subseteq\dots\subseteq I_{1}$.*

Proof. Consider the ordering $a_{1},\dots,a_{k},b_{1},\dots,b_{k},v_{1},\dots,v_{k},v_{k+1}$, and the three classes $A_{k}$, $B_{k}$ and $W_{k}$. It is easy to see that they are strongly consistent.

Let $1\leq i\leq k-1$. Notice that $a_{i}a_{i+1}v_{i+1}b_{i+1}b_{i}$ induce a path of length five on $G_{k}$. In every interval representation of it, the interval $I_{i+1}$ is between the intervals corresponding to $a_{i}$ and $b_{i}$ and disjoint to them. As the five vertices are adjacent to $v_{i}$, it follows that the $I_{i+1}\subseteq I_{i}$. Finally, by the shape of interval representations of a path of length five, each of the intervals corresponding to $a_{k}$ and $b_{k}$ contains an endpoint of $I_{k}$. As $v_{k+1}$ is neither adjacent to $a_{k}$ nor to $b_{k}$, $I_{k+1}\subseteq I_{k}$. $\Box$

The following characterization was proved for $k$-nested interval graphs.

Lemma 13

[24]* An interval graph is $k$-nested interval if and only if it has an interval representation which can be partitioned into $k$ proper interval representations.*

This lemma and the family of graphs $G_{k}$ show that even if the vertices of a proper $k$-thin graph can be partitioned into $k$ sets of vertices each of them inducing a proper interval graph, it is not always the case that it has an interval representation which can be partitioned into $k$ proper interval representations.

6 Solving combinatorial optimization problems on graphs

with bounded thinness

Since a $k$-thin graph $G$ does not contain $\overline{(k+1)K_{2}}$ as induced subgraph (Theorem 1), it has at most $|V(G)|^{2k}$ maximal cliques [33]. In particular, the maximum weighted clique problem can be solved in polynomial time on graphs with bounded thinness, by simple enumeration of the maximal cliques of the graph [41].

The maximum weighted stable set problem can be solved in polynomial time on graphs with bounded thinness, when an ordering and a partition that are consistent are given [29]. In the same hypothesis, the capacitated coloring (in which there is an upper bound $\alpha_{j}$ on the number of vertices of color $j$) can be solved in polynomial time, if the number of colors $s$ is fixed [5]. As a byproduct, in the same paper it is shown that the capacitated coloring can be solved in polynomial time for co-comparability graphs, if the number of colors $s$ is fixed, in contrast with the case in which the bounds $\alpha_{j}$ are all equal to a fixed number $h$, that is NP-complete, even for two subclasses of co-comparability graphs: permutation graphs (for $h\geq 6$) [27] and interval graphs (for $h\geq 4$) [4]. The hardness on interval graphs implies the hardness for graphs of bounded thinness, since interval graphs are the graphs with thinness $1$.

Both algorithms, the one for maximum weighted stable set and the one for capacitated coloring with fixed number of colors, are based on dynamic programming. One of the main results in this work is a generalization of these algorithmic results. We describe now a framework of problems that can be solved for graphs with bounded thinness, given the representation.

Instance:

A $k$-thin representation of $G=(V,E)$, with ordering $<$ of $V$, namely $v_{1}<\dots<v_{n}$, and partition of $V$ into $k$ classes $V^{1},\dots,V^{k}$.

• 2.

A family of arbitrary nonnegative weights $w_{1},\ldots,w_{t}$ on $V$.

• 3.

A family of nonnegative weights $b_{1},\ldots,b_{p}$ on $V$ bounded by a fixed polynomial in $n$ ($p$ fixed, $q(n)$ the bound for the weights).

Question: find sets $S_{1},\dots,S_{r}$ ($r$ fixed, not necessarily disjoint), $S_{j}\subseteq V$ for $1\leq j\leq r$, such that:

the objective is to minimize or maximize a linear function $\sum_{1\leq i\leq t;1\leq j\leq r}c_{ij}w_{i}(S_{j})$.

• 2.

each vertex $v$ has a list $L(v)$ of combinations of the sets $S_{1},\dots,S_{r}$ to which it can belong (that may include the empty combination).

• 3.

there is an $r\times r$ symmetric matrix $M$ over $0,1,*$, stating the adjacency conditions on the sets $S_{j}$, such that for $1\leq i<j\leq r$, $M_{ii}=1$ means $S_{i}$ is a clique, $M_{ii}=0$ means $S_{i}$ is a stable set, $M_{ij}=1$ means all the edges joining $S_{i}$ and $S_{j}$ have to be present, $M_{ij}=0$ means there are no edges from $S_{i}$ to $S_{j}$.

• 4.

there is a family of restrictions on the weight of the intersection and of the union of some families of sets. Such restrictions can be expressed as

• (a)

$0\leq l_{iJ\cap}\leq b_{i}(\bigcap_{j\in J}S_{j})\leq u_{iJ\cap}$, such that $1\leq i\leq p$, $J\subseteq\{1,\dots,r\}$.

• (b)

$0\leq l_{iJ\cup}\leq b_{i}(\bigcup_{j\in J}S_{j})\leq u_{iJ\cup}$, such that $1\leq i\leq p$, $J\subseteq\{1,\dots,r\}$.

Notice that some of these restrictions can be of cardinality, if the corresponding weight function $b_{i}$ is constant.

The family of problems that can be modeled within this framework includes weighted variations of list matrix partition problems with matrices of bounded size, which in turn generalize coloring, list coloring, list homomorphism, equitable coloring with different objective functions, all for fixed number of colors (or graph size in the case of homomorphism), clique cover with fixed number of cliques, weighted stable sets, and other graph partition problems. It models also sum-coloring and its more general version optimum cost chromatic partition problem [22] for fixed number of colors, but it does not include dominating-like problems.

We will solve such a problem as a shortest or longest path problem (according to minimization or maximization of the objective function) in an auxiliary acyclic digraph $D=(X,A)$ whose nodes correspond to states and whose arcs are weighted and labeled. The total weight of the path is the value of the objective function in the solution that can be built by using the arc labels. We will used the term “nodes” for the digraph $D$ in order to avoid confusion with the vertices of the graph $G$.

A state is a tuple, containing:

a number $1\leq s\leq|V_{G}|$ indicating that we are considering the subgraph $G_{s}$ of $G$, induced by $v_{1},\dots,v_{s}$.

• 2.

nonnegative parameters $l_{iJ\cap},u_{iJ\cap},l_{iJ\cup},u_{iJ\cup}$, for $1\leq i\leq p$, $J\subseteq\{1,\dots,r\}$; they are at most $2^{r+2}p$, and each of them may take a nonnegative value at most $nq(n)$, which is an upper bound for $b_{i}(V)$, for every $1\leq i\leq p$.

• 3.

a family of nonnegative parameters $\{\alpha_{ij}\}_{1\leq i\leq k;1\leq j\leq r}$, meaning that we cannot pick for $S_{j}$ a vertex of the first $\alpha_{ij}$ vertices of the set $V^{i}$ of the partition; there are $kr$ such parameters and each of them may take a nonnegative value at most $n-1$.

• 4.

a family of nonnegative parameters $\{\beta_{ij}\}_{1\leq i\leq k;1\leq j\leq r}$, meaning that we cannot pick for $S_{j}$ a vertex on the last $\beta_{ij}$ vertices of the set $V^{i}$ of the partition; there are $kr$ such parameters and each of them may take a nonnegative value at most $n-1$.

The total number of states is then at most $n^{2kr+1}(nq(n))^{2^{r+2}p}$, that is polynomial in $n$, since $k$, $r$, and $p$ are constant and $q(n)$ is polynomial in $n$.

The digraph $D$ will have nodes that correspond to possible states, organized in layers $X_{0},X_{1},\dots,X_{n}$ such that $X_{0}$ contains only one node $x_{0}$, and the layer $X_{s}$ contains the states whose first parameter is $s$. The layer $X_{n}$ contains also only one node, corresponding to the state $(n,$ $\{l_{iJ\cap}\},$ $\{u_{iJ\cap}\},$ $\{l_{iJ\cup}\},$ $\{u_{iJ\cup}\},$ $\{\alpha_{ij}\},$ $\{\beta_{ij}\})$, where the parameters $\{l_{iJ\cap}\},\{u_{iJ\cap}\},\{l_{iJ\cup}\},\{u_{iJ\cup}\}$ are the ones in the original formulation of the problem and $\alpha_{ij}=\beta_{ij}=0$ for every $1\leq i\leq k$, $1\leq j\leq r$.

All arcs of $A$ have the form $(u,w)$ with $u\in X_{s}$ and $w\in X_{s+1}$, for some $0\leq s\leq n-1$.

We associate with each node of $X$ a suitable problem, in the same framework, whose parameters correspond to the parameters in the state, but with additional constraints associated with the parameters $\{\alpha_{ij}\}$ and $\{\beta_{ij}\}$.

We will define the arcs in such a way that a node is reachable from the node in the layer $X_{0}$ if and only if the associated problem has a solution. The length of the path will be the weight of the solution, and the set of arc labels will encode the solution. Let us describe the arcs of the digraph.

Let $w$ be a node with parameters $(1,$ $\{l_{iJ\cap}\},$ $\{u_{iJ\cap}\},$ $\{l_{iJ\cup}\},$ $\{u_{iJ\cup}\},$ $\{\alpha_{ij}\},$ $\{\beta_{ij}\})$.

Let $1\leq\ell\leq k$ such that $v_{1}\in V^{\ell}$. For each $\tilde{J}\in L(v_{1})$ (in particular $\tilde{J}\subseteq\{1,\dots,r\}$), such that:

1. 1.1

For each $j\in\tilde{J}$, $\beta_{\ell j}=\alpha_{\ell j}=0$. 2. 1.2

For each $J\subseteq\tilde{J}$, $l_{iJ\cap}\leq b_{i}(v_{1})\leq u_{iJ\cap}$. 3. 1.3

For each $J\not\subseteq\tilde{J}$, $l_{iJ\cap}=0$. 4. 1.4

For each $J$ such that $J\cap\tilde{J}\neq\emptyset$, $l_{iJ\cup}\leq b_{i}(v_{1})\leq u_{iJ\cup}$. 5. 1.5

For each $J$ such that $J\cap\tilde{J}=\emptyset$, $l_{iJ\cup}=0$.

we add an arc from $x_{0}$ to $w$, labeled by $\tilde{J}$ and of weight $\sum_{1\leq i\leq t;j\in\tilde{J}}c_{ij}w_{i}(v_{1})$. If no $\tilde{J}$ satisfies conditions 1.1–1.5, no arc ending in $w$ is added. If more than one arc $x_{0}w$ was added, we can keep only the one with maximum (resp. minimum) weight if we are solving a maximization (resp. minimization) problem.

Note that if we add the arc $x_{0}w$ labeled by $\tilde{J}$, then the solution $S_{j}=\{v_{1}\}$ for $j\in\tilde{J}$, $S_{j}=\emptyset$ for $j\not\in\tilde{J}$ has weight $\sum_{1\leq i\leq t;j\in\tilde{J}}c_{ij}w_{i}(v_{1})$ and satisfies the state described by $w$: condition 1.1 says that $v_{1}$ (the first and last vertex of $V^{\ell}$ in $G_{1}$) is allowed to be picked for every set $S_{j}$ for $j\in\tilde{J}$; conditions 1.2–1.5 say that the assignment does not violate weight constraints.

Let $w$ be a node with parameters $(s,$ $\{l_{iJ\cap}\},$ $\{u_{iJ\cap}\},$ $\{l_{iJ\cup}\},$ $\{u_{iJ\cup}\},$ $\{\alpha_{ij}\},$ $\{\beta_{ij}\})$, $1<s\leq n$.

Let $1\leq\ell\leq k$ such that $v_{s}\in V^{\ell}$. For each $\tilde{J}\in L(v_{s})$, such that:

1. $s$.1

For each $j\in\tilde{J}$, $\beta_{\ell j}=0$. 2. $s$.2

For each $j\in\tilde{J}$, $\alpha_{\ell j}<|V^{\ell}\cap\{v_{1},\dots,v_{s}\}|$. 3. $s$.3

For each $J\subseteq\tilde{J}$, $b_{i}(v_{s})\leq u_{iJ\cap}$. 4. $s$.4

For each $J$ such that $J\cap\tilde{J}\neq\emptyset$, $b_{i}(v_{s})\leq u_{iJ\cup}$.

we add an arc from $u$ to $w$, labeled by $\tilde{J}$ and of weight $\sum_{1\leq i\leq t;j\in\tilde{J}}c_{ij}w_{i}(v_{s})$, where $u$ has parameters $(s-1,$ $\{l^{\prime}_{iJ\cap}\},$ $\{u^{\prime}_{iJ\cap}\},$ $\{l^{\prime}_{iJ\cup}\},$ $\{u^{\prime}_{iJ\cup}\},$ $\{\alpha^{\prime}_{ij}\},$ $\{\beta^{\prime}_{ij}\})$, such that:

1. $s^{\prime}$.1

Let $1\leq j\leq r$. If there exists $j^{\prime}\in\tilde{J}$ such that $M_{jj^{\prime}}=0$, then $\beta^{\prime}_{\ell j}=\max\{\beta_{\ell j}-1,|N(v_{s})\cap V^{\ell}\cap\{1,\dots,s-1\}|\}$, and for $1\leq i\leq k$, $i\neq\ell$, $\beta^{\prime}_{ij}=\max\{\beta_{ij},|N(v_{s})\cap V^{i}\cap\{1,\dots,s-1\}|\}$. Otherwise, $\beta^{\prime}_{\ell j}=\max\{0,\beta_{\ell j}-1\}$, and for $1\leq i\leq k$, $i\neq\ell$, $\beta^{\prime}_{ij}=\beta_{ij}$. 2. $s^{\prime}$.2

Let $1\leq j\leq r$. If there exists $j^{\prime}\in\tilde{J}$ such that $M_{jj^{\prime}}=1$, then

$\alpha^{\prime}_{\ell j}=\max\{\min\{|V^{\ell}\cap\{1,\dots,s-1\}|,\alpha_{\ell j}\},|\overline{N}(v_{s})\cap V^{\ell}\cap\{1,\dots,s-1\}|\}$, and for $1\leq i\leq k$, $i\neq\ell$, $\alpha^{\prime}_{ij}=\max\{\alpha_{ij},|\overline{N}(v_{s})\cap V^{i}\cap\{1,\dots,s-1\}|\}$. Otherwise, $\alpha^{\prime}_{\ell j}=\min\{|V^{\ell}\cap\{1,\dots,s-1\}|,\alpha_{\ell j}\}$, and for $1\leq i\leq k$, $i\neq\ell$, $\alpha^{\prime}_{ij}=\alpha_{ij}$. 3. $s^{\prime}$.3

For each $J\subseteq\tilde{J}$, $l^{\prime}_{iJ\cap}=\max\{0,l_{iJ\cap}-b_{i}(v_{s})\}$ and $u^{\prime}_{iJ\cap}=u_{iJ\cap}-b_{i}(v_{s})$. 4. $s^{\prime}$.4

For each $J\not\subseteq\tilde{J}$, $l^{\prime}_{iJ\cap}=l_{iJ\cap}$ and $u^{\prime}_{iJ\cap}=u_{iJ\cap}$. 5. $s^{\prime}$.5

For each $J$ such that $J\cap\tilde{J}\neq\emptyset$, $l^{\prime}_{iJ\cup}=\max\{0,l_{iJ\cup}-b_{i}(v_{s})\}$ and $u^{\prime}_{iJ\cup}=u_{iJ\cup}-b_{i}(v_{s})$. 6. $s^{\prime}$.6

For each $J$ such that $J\cup\tilde{J}=\emptyset$, $l^{\prime}_{iJ\cup}=l_{iJ\cup}$ and $u^{\prime}_{iJ\cup}=u_{iJ\cup}$.

If no $\tilde{J}$ satisfies conditions $s$.1–$s$.4, no arc ending in $w$ is added. If more than one arc from the same vertex $u$ to $w$ was added, we can keep only the one with maximum (resp. minimum) weight if we are solving a maximization (resp. minimization) problem.

That is, if an arc is added, the arc corresponds to the choice of adding the vertex $v_{s}$ to the sets $\{S_{j}\}_{j\in\tilde{J}}$, the conditions required imply that the choice is valid for $w$ in the case that the state described by $u$ admits a solution, the label of the arc keeps track of the choice made, and the cost corresponds to the weight that the choice adds to the objective function.

Note that if we add the arc $uw$ labeled by $\tilde{J}$, then for a solution $\{S^{\prime}_{j}\}_{1\leq j\leq r}$ for $G_{s-1}$ satisfying the state described by $u$, then the solution $\{S_{j}\}_{1\leq j\leq r}$ for $G_{s}$ such that $S_{j}=S^{\prime}_{j}\cup\{v_{s}\}$ for $j\in\tilde{J}$, $S_{j}=S^{\prime}_{j}$ for $j\not\in\tilde{J}$ satisfies the state described by $w$. Conditions $s$.1 and $s$.2 say that $v_{s}$ (the last vertex of $V^{\ell}$ in $G_{s}$) is allowed to be picked for every set $S_{j}$ for $j\in\tilde{J}$. Condition $s^{\prime}$.1 ensures on one hand that the conditions imposed by the parameters $\{\beta_{ij}\}$ in $w$ are satisfied by the solution of $u$, and, on the other hand, that if $j^{\prime}\in\tilde{J}$ and $1\leq j\leq r$ are such that $M_{jj^{\prime}}=0$ then no neighbor of $v_{s}$ belongs to $S^{\prime}_{j}$, as required. Similarly, condition $s^{\prime}$.2 ensures on one hand that the conditions imposed by the parameters $\{\alpha_{ij}\}$ in $w$ are satisfied also by the solution of $u$, and, on the other hand, that if $j^{\prime}\in\tilde{J}$ and $1\leq j\leq r$ are such that $M_{jj^{\prime}}=1$ then all vertices in $S^{\prime}_{j}$ are adjacent to $v_{s}$, as required. These conditions strongly use that the order and the partition are consistent. Finally, conditions $s$.3–$s$.4, and $s^{\prime}$.3–$s^{\prime}$.6 ensure that the solution does not violate weight constraints.

Moreover, the difference of weight of the solution $\{S_{j}\}_{1\leq j\leq r}$ with respect to $\{S^{\prime}_{j}\}_{1\leq j\leq r}$ is exactly $\sum_{1\leq i\leq t;j\in\tilde{J}}c_{ij}w_{i}(v_{s})$.

In that way, a directed path in the digraph corresponds to an assignment of vertices of the graph to lists of sets and its weight is the value of the objective function for the corresponding assignment.

The digraph has a polynomial number of nodes and can be built in polynomial time. Since it is acyclic, both the longest path and shortest path can be computed in linear time in the size of the digraph by topological sorting.

Remark 1

The thinness is not preserved by the complement operation of graphs (see for instance Theorem 1). However, for every fixed $k$, all the problems that can be modeled in this framework can be solved for the complement $\overline{G}$ of a $k$-thin graph $G$, in the same framework, simply by swapping ones and zeroes in the restriction matrix $M$.

6.1 Extending the family of combinatorial optimization problems solvable on graphs with bounded proper

thinness

We start by the following observation: in a proper $k$-thin representation of a graph $G$, with ordering $<$ of $V$, namely $v_{1}<\dots<v_{n}$, and partition of $V$ into $k$ classes $V^{1},\dots,V^{k}$, for each pair of vertices $v_{s}<v_{r}$ that are in the same class, $N[v_{s}]\cap\{v_{1},\dots,v_{s}\}\supseteq N[v_{r}]\cap\{v_{1},\dots,v_{s}\}$. This allows us to handle other kinds of restrictions as for example domination type constraints.

Namely, if we are considering the subgraph $G_{s}$ of $G$ induced by $\{v_{1},\dots,v_{s}\}$ but we “keep in mind” that we still need to dominate some of the vertices in $\{v_{s+1},\dots,v_{n}\}$ with vertices of $G_{s}$, we can summarize these conditions into at most $k$ of them (each imposed by vertices of $\{v_{s+1},\dots,v_{n}\}$ in each partition class).

For graphs with bounded proper thinness $k$, when the proper $k$-thin representation of the graph is given, we can add now to the instance (with respect to Section 6) this kind of restrictions:

$l_{ij(N)}\leq|S_{i}\cap N(v)|\leq u_{ij(N)}\quad\forall v\in S_{j}$, such that $l_{ij(N)}\in\{0,1\}$ and $u_{ij(N)}\in\{1,\infty\}$ (it can be $i=j$), $1\leq i,j\leq r$.

• 2.

$l_{ij[N]}\leq|S_{i}\cap N[v]|\leq u_{ij[N]}\quad\forall v\in S_{j}$, such that $l_{ij[N]}\in\{0,1\}$ and $u_{ij[N]}\in\{1,\infty\}$ (it can be $i=j$), $1\leq i,j\leq r$.

In this way the framework includes domination-type problems in the literature and their weighted versions, such as existence/minimum (weighted) independent dominating set, minimum (weighted) total dominating set, minimum perfect dominating set and existence/minimum (weighted) efficient dominating set, b-coloring [21] with fixed number of colors.

We will keep the notation of Section 6 and describe how to modify the algorithm in order to take into account the new restrictions. Now the vertex order and the partition of $G$ are strongly consistent.

Each state now will be augmented with some new parameters:

a family of nonnegative parameters $\{\gamma_{ij}\}_{1\leq i\leq k;1\leq j\leq r}$, meaning that the last $\gamma_{ij}$ vertices of $V^{i}$ have already a neighbor in $S_{j}$ (of index higher than them); there are $kr$ such parameters and each of them may take a nonnegative value at most $n-1$.

• 2.

a family of nonnegative parameters $\{\gamma^{2}_{ij}\}_{1\leq i\leq k;1\leq j\leq r}$, meaning that the last $\gamma^{2}_{ij}$ vertices of $V^{i}$ have already two neighbors in $S_{j}$ (of index higher than them); there are $kr$ such parameters and each of them may take a nonnegative value at most $n-1$.

• 3.

a family of nonnegative parameters $\{\lambda_{ijc}\}_{1\leq i,c\leq k;1\leq j\leq r}$, meaning that, for each value $1\leq c\leq k$, $S_{j}$ has to contain at least one vertex in the set that is the union over $1\leq i\leq k$ of the last $\lambda_{ijc}$ vertices of $V^{i}$ (if the union is empty, this means no restriction associated with $(c,S_{j})$); there are $k^{2}r$ such parameters and each of them may take a nonnegative value at most $n-1$.

The total number of states is then multiplied by at most $n^{k^{2}r+2kr}$, that keeps it polynomial in $n$, since $k$ and $r$ are constant.

The value of all these parameters in the only node of the layer $X_{n}$ of the digraph is zero.

Now the problems associated with the nodes of $X$ will have the additional constraints associated with the new restrictions and the parameters $\{\gamma_{ij}\}$, $\{\gamma^{2}_{ij}\}$, and $\{\lambda_{ijc}\}$.

Let us describe the additional conditions for the arcs of the digraph, whose labels and weights are still the same as in Section 6.

Let $w$ be a node with parameters $(1,$ $\dots,$ $\{\gamma_{ij}\},$ $\{\gamma^{2}_{ij}\},$ $\{\lambda_{ijc}\})$.

Let $1\leq\ell\leq k$ such that $v_{1}\in V^{\ell}$. For each $\tilde{J}\in L(v_{1})$ (in particular $\tilde{J}\subseteq\{1,\dots,r\}$) satisfying 1.1–1.5, and such that:

1. 1.6

For each $1\leq i\leq r$, $j\in\tilde{J}$, such that $l_{ij(N)}=1$, $\gamma_{\ell i}>0$. 2. 1.7

For each $i\not\in\tilde{J}$, $j\in\tilde{J}$, such that $l_{ij[N]}=1$, $\gamma_{\ell i}>0$. 3. 1.8

For each $1\leq i\leq r$, $j\in\tilde{J}$, such that $u_{ij(N)}=1$ or $u_{ij[N]}=1$, $\gamma^{2}_{\ell i}=0$. 4. 1.9

For each $i,j\in\tilde{J}$, such that $u_{ij[N]}=1$, $\gamma_{\ell i}=0$. 5. 1.10

For each $j\not\in\tilde{J}$ and for each $1\leq c\leq k$, $\lambda_{\ell jc}=0$.

we add an arc from $x_{0}$ to $w$, labeled by $\tilde{J}$ and of weight $\sum_{1\leq i\leq t;j\in\tilde{J}}c_{ij}w_{i}(v_{1})$. If no $\tilde{J}$ satisfies conditions 1.1–1.10, no arc ending in $w$ is added. If more than one arc $x_{0}w$ was added, we can keep only the one with maximum (resp. minimum) weight if we are solving a maximization (resp. minimization) problem.

Note that if we add the arc $x_{0}w$ labeled by $\tilde{J}$, then the solution $S_{j}=\{v_{1}\}$ for $j\in\tilde{J}$, $S_{j}=\emptyset$ for $j\not\in\tilde{J}$ has weight $\sum_{1\leq i\leq t;j\in\tilde{J}}c_{ij}w_{i}(v_{1})$ and satisfies the state described by $w$: conditions 1.1–1.5 ensure the properties required in Section 6; conditions 1.6–1.9 ensure the validity of the two new families of restrictions about lower and upper bounds of neighbors of vertices of one set in other set, and condition 1.10 ensures that the restrictions imposed by the parameters $\{\lambda_{ijc}\}$ are satisfied.

Let $w$ be a node with parameters $(s,$ $\{l_{iJ\cap}\},$ $\{u_{iJ\cap}\},$ $\{l_{iJ\cup}\},$ $\{u_{iJ\cup}\},$ $\{\alpha_{ij}\},$ $\{\beta_{ij}\}$, $\{\gamma_{ij}\},$ $\{\gamma^{2}_{ij}\},$ $\{\lambda_{ijc}\})$, $1<s\leq n$.

Let $1\leq\ell\leq k$ such that $v_{s}\in V^{\ell}$. For each $\tilde{J}\in L(v_{s})$ satisfying $s$.1–$s$.4, and such that:

1. $s$.5

For each $1\leq i\leq r$, $j\in\tilde{J}$, such that $u_{ij(N)}=1$ or $u_{ij[N]}=1$, $\gamma^{2}_{\ell i}=0$. 2. $s$.6

For each $i,j\in\tilde{J}$, such that $u_{ij[N]}=1$, $\gamma_{\ell i}=0$. 3. $s$.7

For each $j\not\in\tilde{J}$ and for each $1\leq c\leq k$, either $\lambda_{\ell jc}=0$, or $\lambda_{\ell jc}>1$, or there exists $1\leq i\leq k$, $i\neq\ell$, such that $\lambda_{ijc}>0$ (i.e., the union over $1\leq i\leq k$ of the last $\lambda_{ijc}$ vertices of $V^{i}$ is not $\{v_{s}\}$). 4. $s$.8

For each $1\leq i\leq r$ such that $\gamma_{\ell i}=0$ and there exists $j\in\tilde{J}$ such that $l_{ij(N)}=1$, $N(v_{s})\cap\{1,\dots,s-1\}\neq\emptyset$. 5. $s$.9

For each $i\not\in\tilde{J}$ such that $\gamma_{\ell i}=0$ and there exists $j\in\tilde{J}$ such that $l_{ij[N]}=1$, $N(v_{s})\cap\{1,\dots,s-1\}\neq\emptyset$.

Let $\{\lambda^{0}_{ijc}\}_{1\leq i,c\leq k;1\leq j\leq r}$ be defined this way: for every $j\in\tilde{J}$ and every $1\leq c\leq k$ such that $\lambda_{\ell jc}>0$, let $\lambda^{0}_{ijc}=0$ for every $1\leq i\leq k$; for every $j\in\tilde{J}$ and every $1\leq c\leq k$ such that $\lambda_{\ell jc}=0$, let $\lambda^{0}_{ijc}=\lambda_{ijc}$ for every $1\leq i\leq k$; for every $j\not\in\tilde{J}$ and every $1\leq c\leq k$, let $\lambda^{0}_{\ell jc}=\max\{0,\lambda_{\ell jc}-1\}$ and let $\lambda^{0}_{ijc}=\lambda_{ijc}$ for every $1\leq i\leq k$, $i\neq\ell$.

Let $\{\lambda^{1}_{jc}\}_{1\leq c\leq k;1\leq j\leq r}$ be defined as $\lambda^{1}_{jc}=0$ if $\lambda^{0}_{ijc}=0$ for every $1\leq i\leq k$, $\lambda^{1}_{jc}=1$ otherwise.

We add an arc from $u$ to $w$, labeled by $\tilde{J}$ and of weight $\sum_{1\leq i\leq t;j\in\tilde{J}}c_{ij}w_{i}(v_{s})$, where $u$ has parameters $(s-1,$ $\{l^{\prime}_{iJ\cap}\},$ $\{u^{\prime}_{iJ\cap}\},$ $\{l^{\prime}_{iJ\cup}\},$ $\{u^{\prime}_{iJ\cup}\},$ $\{\alpha^{\prime}_{ij}\},$ $\{\beta^{\prime}_{ij}\}$, $\{\gamma^{\prime}_{ij}\},$ $\{{\gamma^{2}}^{\prime}_{ij}\},$ $\{\lambda^{\prime}_{ijc}\})$, satisfies conditions $s^{\prime}$.2–$s^{\prime}$.6, and:

1. $s^{\prime}$.7

For each $1\leq i\leq r$ such that $\gamma_{\ell i}=0$ and there exists $j\in\tilde{J}$ such that $l_{ij(N)}=1$, if $\lambda^{1}_{i\ell}=0$, then $\lambda^{\prime}_{j^{\prime}i\ell}=|N(v_{s})\cap V^{j^{\prime}}\cap\{1,\dots,s-1\}|$ for each $1\leq j^{\prime}\leq k$; otherwise, $\lambda^{\prime}_{j^{\prime}i\ell}=\lambda^{0}_{j^{\prime}i\ell}$ for every $1\leq j^{\prime}\leq k$ (recall that, by the observations above about proper thinness, $\lambda^{0}_{j^{\prime}i\ell}=\min\{\lambda^{0}_{j^{\prime}i\ell},|N(v_{s})\cap V^{j^{\prime}}\cap\{1,\dots,s-1\}|\}$). 2. $s^{\prime}$.8

For each $i\not\in\tilde{J}$ such that $\gamma_{\ell i}=0$ and there exists $j\in\tilde{J}$ such that $l_{ij[N]}=1$, if $\lambda^{1}_{i\ell}=0$, then $\lambda^{\prime}_{j^{\prime}i\ell}=|N(v_{s})\cap V^{j^{\prime}}\cap\{1,\dots,s-1\}|$ for each $1\leq j^{\prime}\leq k$; otherwise, $\lambda^{\prime}_{j^{\prime}i\ell}=\lambda^{0}_{j^{\prime}i\ell}$ for every $1\leq j^{\prime}\leq k$. 3. $s^{\prime}$.9

For each $i,j,c$ not comprised in conditions $s^{\prime}$.7 and $s^{\prime}$.8, $\lambda^{\prime}_{ijc}=\lambda^{0}_{ijc}$. 4. $s^{\prime}$.10

Let $1\leq j\leq r$. If there exists $j^{\prime}\in\tilde{J}$ satisfying at least one of the following:

• (a)

$M_{jj^{\prime}}=0$

• (b)

($u_{jj^{\prime}(N)}=1$ or $u_{jj^{\prime}[N]}=1$) and $\gamma_{\ell j}>0$

• (c)

$j\in\tilde{J}$ and $u_{jj^{\prime}[N]}=1$

then, $\beta^{\prime}_{\ell j}=\max\{\beta_{\ell j}-1,|N(v_{s})\cap V^{\ell}\cap\{1,\dots,s-1\}|\}$, and for $1\leq i\leq k$, $i\neq\ell$, $\beta^{\prime}_{ij}=\max\{\beta_{ij},|N(v_{s})\cap V^{i}\cap\{1,\dots,s-1\}|\}$. Otherwise, $\beta^{\prime}_{\ell j}=\max\{0,\beta_{\ell j}-1\}$, and for $1\leq i\leq k$, $i\neq\ell$, $\beta^{\prime}_{ij}=\beta_{ij}$. 5. $s^{\prime}$.11

For each $j\in\tilde{J}$: if $|N(v_{s})\cap V^{\ell}\cap\{1,\dots,s-1\}|\geq\gamma_{\ell j}-1$, then $\gamma^{\prime}_{\ell j}=|N(v_{s})\cap V^{\ell}\cap\{1,\dots,s-1\}|$ and ${\gamma^{2}}^{\prime}_{\ell j}=\max\{0,\gamma_{\ell j}-1\}$; otherwise, $\gamma^{\prime}_{\ell j}=\max\{0,\gamma_{\ell j}-1\}$ and ${\gamma^{2}}^{\prime}_{\ell j}=\max\{\gamma^{2}_{\ell j}-1,|N(v_{s})\cap V^{\ell}\cap\{1,\dots,s-1\}|\}$. 6. $s^{\prime}$.12

For each $j\in\tilde{J}$, $1\leq i\leq k$, $i\neq\ell$: if $|N(v_{s})\cap V^{i}\cap\{1,\dots,s-1\}|\geq\gamma_{ij}$, then $\gamma^{\prime}_{ij}=|N(v_{s})\cap V^{i}\cap\{1,\dots,s-1\}|$ and ${\gamma^{2}}^{\prime}_{ij}=\gamma_{ij}$; otherwise, $\gamma^{\prime}_{ij}=\gamma_{ij}$ and ${\gamma^{2}}^{\prime}_{ij}=\max\{\gamma^{2}_{ij},|N(v_{s})\cap V^{i}\cap\{1,\dots,s-1\}|\}$.

If no $\tilde{J}$ satisfies conditions $s$.1–$s$.9, no arc ending in $w$ is added. If more than one arc from the same vertex $u$ to $w$ was added, we can keep only the one with maximum (resp. minimum) weight if we are solving a maximization (resp. minimization) problem.

That is, if an arc is added, the arc corresponds to the choice of adding the vertex $v_{s}$ to the sets $\{S_{j}\}_{j\in\tilde{J}}$, the conditions required imply that the choice is valid for $w$ in the case that the state described by $u$ admits a solution, the label of the arc keeps track of the choice made, and the cost corresponds to the weight that the choice adds to the objective function.

Note that if we add the arc $uw$ labeled by $\tilde{J}$, then for a solution $\{S^{\prime}_{j}\}_{1\leq j\leq r}$ for $G_{s-1}$ satisfying the state described by $u$, then the solution $\{S_{j}\}_{1\leq j\leq r}$ for $G_{s}$ such that $S_{j}=S^{\prime}_{j}\cup\{v_{s}\}$ for $j\in\tilde{J}$, $S_{j}=S^{\prime}_{j}$ for $j\not\in\tilde{J}$ satisfies the state described by $w$.

Condition $s^{\prime}$.10 ensures on one hand that the conditions imposed by the parameters $\{\beta_{ij},u_{ij(N)},u_{ij[N]}\}$ in $w$ are satisfied by the solution of $u$, and, on the other hand, that if $j^{\prime}\in\tilde{J}$ and $1\leq j\leq r$ are such that $M_{jj^{\prime}}=0$ then no neighbor of $v_{s}$ belongs to $S^{\prime}_{j}$, as required. Conditions $s^{\prime}$.7–$s^{\prime}$.9 together with $s$.7–$s$.9 define parameters $\{\lambda^{\prime}_{ijc}\}$ in $u$ in order to guarantee in $w$ both the conditions imposed by the lower bounds $\{l_{ij(N)},l_{ij[N]}\}$ and those imposed by the parameters $\{\lambda_{ijc}\}$. Finally, conditions $s^{\prime}$.11 and $s^{\prime}$.12 properly update the definition of parameters $\{\gamma^{\prime}_{ij},{\gamma^{2}}^{\prime}_{ij}\}$ according to the choice $\tilde{J}$ for $v_{s}$. Conditions $s^{\prime}$.2–$s^{\prime}$.6 were analyzed above in Section 6.

As in that case, the difference of weight of the solution $\{S_{j}\}_{1\leq j\leq r}$ with respect to $\{S^{\prime}_{j}\}_{1\leq j\leq r}$ is exactly $\sum_{1\leq i\leq t;j\in\tilde{J}}c_{ij}w_{i}(v_{s})$.

In that way, a directed path in the digraph corresponds to an assignment of vertices of the graph to lists of sets and its weight is the value of the objective function for the corresponding assignment.

The digraph has a polynomial number of nodes and can be built in polynomial time. Since it is acyclic, both the longest path and shortest path can be computed in linear time in the size of the digraph by topological sorting.

7 Thinness and graph operations

In this section we analyze the behavior of the thinness and proper thinness under different graph operations, namely union, join, and Cartesian product. The first two results allow us to fully characterize $k$-thin graphs by forbidden induced subgraphs within the class of cographs. The third result is used to solve in polynomial time the $t$-rainbow domination problem for fixed $t$ on graphs with bounded thinness.

Let $G_{1}=(V_{1},E_{1})$ and $G_{2}=(V_{2},E_{2})$ be two graphs with $V_{1}\cap V_{2}=\emptyset$. The union of $G_{1}$ and $G_{2}$ is the graph $G_{1}\cup G_{2}=(V_{1}\cup V_{2},E_{1}\cup E_{2})$, and the join of $G_{1}$ and $G_{2}$ is the graph $G_{1}\vee G_{2}=(V_{1}\cup V_{2},E_{1}\cup E_{2}\cup V_{1}\times V_{2})$ (i.e., $\overline{G_{1}\vee G_{2}}=\overline{G_{1}}\cup\overline{G_{2}}$).

Theorem 14

Let $G_{1}$ and $G_{2}$ be graphs. Then $\operatorname{thin}(G_{1}\cup G_{2})=\max\{\operatorname{thin}(G_{1}),$ $\operatorname{thin}(G_{2})\}$ and $\operatorname{pthin}(G_{1}\cup G_{2})=\max\{\operatorname{pthin}(G_{1}),\operatorname{pthin}(G_{2})\}$.

Proof. Since both $G_{1}$ and $G_{2}$ are induced subgraphs of $G_{1}\cup G_{2}$, then $\operatorname{thin}(G_{1}\cup G_{2})\geq\max\{\operatorname{thin}(G_{1}),\operatorname{thin}(G_{2})\}$ and the same holds for the proper thinness.

Let $G_{1}$ and $G_{2}$ be two graphs with thinness (resp. proper thinness) $t_{1}$ and $t_{2}$, respectively. Let $v_{1},\dots,v_{n_{1}}$ and $(V_{1}^{1},\ldots,V_{1}^{t_{1}})$ be an ordering and a partition of $V(G_{1})$ which are consistent (resp. strongly consistent). Let $w_{1},\dots,w_{n_{2}}$ and $(V_{2}^{1},\ldots,V_{2}^{t_{2}})$ be an ordering and a partition of $V(G_{2})$ which are consistent (resp. strongly consistent). Suppose without loss of generality that $t_{1}\leq t_{2}$. For $G=G_{1}\cup G_{2}$, define a partition $V^{1},\dots,V^{t_{2}}$ such that $V^{i}=V_{1}^{i}\cup V_{2}^{i}$ for $i=1,\dots,t_{1}$ and $V^{i}=V_{2}^{i}$ for $i=t_{1}+1,\dots,t_{2}$, and define $v_{1},\dots,v_{n_{1}},w_{1},\dots,w_{n_{2}}$ as an ordering of the vertices. By definition of union of graphs, if three ordered vertices according to the order defined in $V(G_{1}\cup G_{2})$ are such that the first and the third are adjacent, either the three vertices belong to $V(G_{1})$ or the three vertices belong to $V(G_{2})$. Since the order and the partition restricted to each of $G_{1}$ and $G_{2}$ are the original ones, the properties required for consistency (resp. strong consistency) are satisfied. $\Box$

Theorem 15

Let $G_{1}$ and $G_{2}$ be graphs. Then $\operatorname{thin}(G_{1}\vee G_{2})\leq\operatorname{thin}(G_{1})+\operatorname{thin}(G_{2})$ and $\operatorname{pthin}(G_{1}\vee G_{2})\leq\operatorname{pthin}(G_{1})+\operatorname{pthin}(G_{2})$. Moreover, if $G_{2}$ is complete, then $\operatorname{thin}(G_{1}\vee G_{2})=\operatorname{thin}(G_{1})$.

Proof. Let $G_{1}$ and $G_{2}$ be two graphs with thinness (resp. proper thinness) $t_{1}$ and $t_{2}$, respectively. Let $v_{1},\dots,v_{n_{1}}$ and $(V_{1}^{1},\ldots,V_{1}^{t_{1}})$ be an ordering and a partition of $V(G_{1})$ which are consistent (resp. strongly consistent). Let $w_{1},\dots,w_{n_{2}}$ and $(V_{2}^{1},\ldots,V_{2}^{t_{2}})$ be an ordering and a partition of $V(G_{2})$ which are consistent (resp. strongly consistent). For $G=G_{1}\vee G_{2}$, define a partition with $t_{1}+t_{2}$ sets as the union of the two partitions, and $v_{1},\dots,v_{n_{1}},w_{1},\dots,w_{n_{2}}$ as an ordering of the vertices.

Let $x,y,z$ be three vertices of $V(G)$ such that $x<y<z$, $xz\in E(G)$, and $x$ and $y$ are in the same class of the partition of $V(G)$. Then, in particular, $x$ and $y$ both belong either to $V(G_{1})$ or to $V(G_{2})$. If $z$ belongs to the same graph, then $yz\in E(G)$ because the ordering and partition restricted to each of $G_{1}$ and $G_{2}$ are consistent. Otherwise, $z$ is also adjacent to $y$ by the definition of join.

We have proved that the defined partition and ordering are consistent, and thus that $\operatorname{thin}(G_{1}\vee G_{2})\leq\operatorname{thin}(G_{1})+\operatorname{thin}(G_{2})$. The proof of the strong consistency, given the strong consistency of the partition and ordering of each of $G_{1}$ and $G_{2}$, is symmetric and implies $\operatorname{pthin}(G_{1}\vee G_{2})\leq\operatorname{pthin}(G_{1})+\operatorname{pthin}(G_{2})$.

Suppose now that $G_{2}$ is complete (in particular, $t_{2}=1$). Since $G_{1}$ is an induced subgraph of $G_{1}\vee G_{2}$, then $\operatorname{thin}(G_{1}\vee G_{2})\geq\operatorname{thin}(G_{1})$. For $G=G_{1}\vee G_{2}$, define a partition $V^{1},\dots,V^{t_{1}}$ such that $V^{1}=V_{1}^{1}\cup V_{2}^{1}$ and $V^{i}=V_{1}^{i}$ for $i=2,\dots,t_{1}$, and define $v_{1},\dots,v_{n_{1}},w_{1},\dots,w_{n_{2}}$ as an ordering of the vertices.

Let $x,y,z$ be three vertices of $V(G)$ such that $x<y<z$, $xz\in E(G)$, and $x$ and $y$ are in the same class of the partition of $V(G)$. If $z$ belongs to $V(G_{2})$, then $z$ is also adjacent to $y$, because it is adjacent to every vertex in $G-z$. If $z$ belongs to $V(G_{1})$, then $x$, $y$, and $z$, belong to $V(G_{1})$ due to the definition of the order of the vertices, and thus $yz\in E(G)$ because the ordering and partition restricted to $G_{1}$ are consistent. This proves $\operatorname{thin}(G_{1}\vee G_{2})\leq\operatorname{thin}(G_{1})$, and therefore $\operatorname{thin}(G_{1}\vee G_{2})=\operatorname{thin}(G_{1})$. $\Box$

The following lemma shows a way of obtaining graphs with high thinness, using the join operator.

Lemma 16

If $G$ is not complete, then $\operatorname{thin}(G\vee 2K_{1})=\operatorname{thin}(G)+1$.

Proof. By Theorem 15, $\operatorname{thin}(G\vee 2K_{1})\leq\operatorname{thin}(G)+\operatorname{thin}(2K_{1})=\operatorname{thin}(G)+1$. On the other hand, as $G\vee 2K_{1}$ contains $G$ as induced subgraph, $\operatorname{thin}(G\vee 2K_{1})\geq\operatorname{thin}(G)$.

First notice that if $\operatorname{thin}(G)=1$ but $G$ is not complete, then $G\vee 2K_{1}$ contains $C_{4}$ as induced subgraph, so it is not an interval graph, and $\operatorname{thin}(G\vee 2K_{1})\geq 2$, as claimed.

Suppose then that $\operatorname{thin}(G)=k>1$ and $\operatorname{thin}(G\vee 2K_{1})=k$ as well, and let $<$ be an ordering of the vertices of $G\vee 2K_{1}$ consistent with a partition $V^{1},\dots,V^{k}$. Let $v,v^{\prime}$ be the vertices of the graph $2K_{1}$, and suppose $v<v^{\prime}$. Without loss of generality we may assume $v\in V^{k}$. As $\operatorname{thin}(G)=k$, $V^{k}\cap V(G)\neq\emptyset$. Since $v^{\prime}>v$, $v^{\prime}$ is nonadjacent to $v$, and $v^{\prime}$ is adjacent to all the vertices in $V^{k}\cap V(G)$, $v$ has to be the smallest vertex in $V^{k}$. Let $z\in V^{k}\cap V(G)$ and suppose there is a vertex $x>z$ in $V(G)$. As $x$ is adjacent to $v^{\prime}$, it is adjacent to $z$ as well. So, we can define a new order $<^{\prime}$ on $V(G\vee 2K_{1})$ that preserves the order $<$ in $V^{1}\cup V^{k-1}\cup\{v\}$ and such that the vertices of $V^{k}-\{v\}$ are the largest. By the observations above, this order $<^{\prime}$ is still consistent with the partition $V^{1},\dots,V^{k}$. But it is also consistent with the partition ${V^{1}}^{\prime},\dots,{V^{k}}^{\prime}$ in which ${V^{1}}^{\prime}=V^{1}\cup V^{k}-\{v\}$, ${V^{i}}^{\prime}=V^{i}$ for $1<i<k$, and ${V^{k}}^{\prime}=\{v\}$. This implies that $\operatorname{thin}(G)<k$, a contradiction that completes the proof of the theorem. $\Box$

Cographs were defined in [9], where it was shown that they are exactly the graphs with no induced path of length four. Cographs admit a full decomposition theorem. Let the trivial graph be the one with one vertex only.

Proposition 17

[9]* Every cograph that is not trivial is either the union or the join of two smaller cographs.*

We will use this structural property along with the theorems about thinness of union and join of graphs to prove the following.

Theorem 18

Let $G$ be a cograph and $t\geq 1$. Then $G$ has thinness at most $t$ if and only if $G$ contains no $\overline{(t+1)K_{2}}$ as induced subgraph.

Proof. The only if part holds by Theorem 1, because the class of $k$-thin graph is hereditary for every $k$.

We will prove the if part by induction on the number of vertices of the cograph $G$. If $G$ is a trivial graph, then $\operatorname{thin}(G)=1$ and the theorem holds. If $G$ is not trivial, by Proposition 17, it is either union or join of two smaller cographs $G_{1}$ and $G_{2}$, with thinness $t_{1}$ and $t_{2}$, respectively.

Suppose first $G=G_{1}\cup G_{2}$. By Theorem 14, $\operatorname{thin}(G)=\max\{t_{1},t_{2}\}$. If $t_{1}$ (resp. $t_{2}$) is greater than one, then by inductive hypothesis $G_{1}$ (resp. $G_{2}$) contains $\overline{t_{1}K_{2}}$ (resp. $\overline{t_{2}K_{2}}$) as induced subgraph, and so does $G$.

Suppose now that $G=G_{1}\vee G_{2}$. If one of them is complete (suppose without loss of generality $G_{2}$), then, by Theorem 15, $\operatorname{thin}(G)=t_{1}$. If $t_{1}$ is greater than one, then by inductive hypothesis $G_{1}$ contains $\overline{t_{1}K_{2}}$ as induced subgraph, and so does $G$. If none of them is complete, then, by that fact and the inductive hypothesis, $G_{1}$ contains $\overline{t_{1}K_{2}}$ and $G_{2}$ contains $\overline{t_{2}K_{2}}$ as induced subgraph. As $\overline{t_{1}K_{2}}\vee\overline{t_{2}K_{2}}=\overline{(t_{1}+t_{2})K_{2}}$, $G$ contains $\overline{(t_{1}+t_{2})K_{2}}$ as induced subgraph, thus $\operatorname{thin}(G)\geq t1+t2$ (Theorem 1). By Theorem 15, $\operatorname{thin}(G)\leq t1+t2$, and therefore $\operatorname{thin}(G)=t1+t2$. This finishes the proof of the theorem. $\Box$

A characterization by minimal forbidden induced subgraphs for $k$-thin graphs, $k\geq 2$, is open.

Let $G_{1}=(V_{1},E_{1})$ and $G_{2}=(V_{2},E_{2})$ be two graphs. The Cartesian product $G_{1}\square G_{2}$ is a graph whose vertex set is the Cartesian product $V_{1}\times V_{2}$, and such that two vertices $(u_{1},u_{2})$ and $(v_{1},v_{2})$ are adjacent in $G_{1}\square G_{2}$ if and only if either $u_{1}=v_{1}$ and $u_{2}$ is adjacent to $v_{2}$ in $G_{2}$, or $u_{2}=v_{2}$ and $u_{1}$ is adjacent to $v_{1}$ in $G_{1}$.

Theorem 19

Let $G_{1}$ and $G_{2}$ be graphs. Then $\operatorname{thin}(G_{1}\square G_{2})\leq\operatorname{thin}(G_{1})|V(G_{2})|$ and $\operatorname{pthin}(G_{1}\square G_{2})\leq\operatorname{pthin}(G_{1})|V(G_{2})|$.

Proof. Let $G_{1}=(V_{1},E_{1})$ be a $k$-thin (resp. proper $k$-thin) graph, and let $v_{1},\dots,v_{n_{1}}$ and $(V_{1}^{1},\ldots,V_{1}^{k})$ be an ordering and a partition of $V_{1}$ which are consistent (resp. strongly consistent). Let $G_{2}=(V_{2},E_{2})$, $n_{2}=|V_{2}|$, and $w_{1},\dots,w_{n_{2}}$ an arbitrary ordering of $V_{2}$. Consider $V_{1}\times V_{2}$ lexicographically ordered with respect to the orderings of $V_{1}$ and $V_{2}$ above. Consider now the partition $\{V^{i,j}\}_{1\leq i\leq k,\ 1\leq j\leq n_{2}}$ such that $V^{i,j}=\{(v,w_{j}):v\in V_{1}^{i}\}$ for each $1\leq i\leq k$, $1\leq j\leq n_{2}$. We will show that this ordering and partition of $V_{1}\times V_{2}$ are consistent (resp. strongly consistent). Let $(v_{p},w_{i}),(v_{q},w_{j}),(v_{r},w_{\ell})$ be three vertices appearing in that ordering in $V_{1}\times V_{2}$.

Case 1: $p=q=r$. In this case, the three vertices are in different classes, so no restriction has to be satisfied.

Case 2: $p=q<r$. In this case, $(v_{p},w_{i})$ and $(v_{q},w_{j})$ are in different classes. So suppose $G_{1}$ is proper $k$-thin and $(v_{q},w_{j}),(v_{r},w_{\ell})$ belong to the same class, i.e., $j=\ell$. Vertices $(v_{p},w_{i})$ and $(v_{r},w_{\ell})$ are adjacent in $G_{1}\square G_{2}$ if and only if $i=\ell$ and $v_{p}v_{r}\in E_{1}$. But then $(v_{p},w_{i})=(v_{q},w_{j})$, a contradiction.

Case 3: $p<q=r$. In this case, $(v_{q},w_{j})$ and $(v_{r},w_{\ell})$ are in different classes. So suppose $G_{1}$ is $k$-thin and $(v_{p},w_{i}),(v_{q},w_{j})$ belong to the same class, i.e., $i=j$. Vertices $(v_{p},w_{i})$ and $(v_{r},w_{\ell})$ are adjacent in $G_{1}\square G_{2}$ if and only if $i=\ell$ and $v_{p}v_{r}\in E_{1}$. But then $(v_{r},w_{\ell})=(v_{q},w_{j})$, a contradiction.

Case 4: $p<q<r$. Suppose first $G_{1}$ is $k$-thin (resp. proper $k$-thin) and $(v_{p},w_{i}),(v_{q},w_{j})$ belong to the same class, i.e., $i=j$ and $v_{p}$, $v_{q}$ belong to the same class in $G_{1}$. Vertices $(v_{p},w_{i})$ and $(v_{r},w_{\ell})$ are adjacent in $G_{1}\square G_{2}$ if and only if $i=\ell$ and $v_{p}v_{r}\in E_{1}$. But then $j=\ell$ and since the ordering and the partition are consistent (resp. strongly consistent) in $G_{1}$, $v_{r}v_{q}\in E_{1}$ and so $(v_{r},w_{\ell})$ and $(v_{q},w_{j})$ are adjacent in $G_{1}\square G_{2}$. Now suppose that $G_{1}$ is proper $k$-thin and $(v_{q},w_{j}),(v_{r},w_{\ell})$ belong to the same class, i.e., $j=\ell$. Vertices $(v_{p},w_{i})$ and $(v_{r},w_{\ell})$ are adjacent in $G_{1}\square G_{2}$ if and only if $i=\ell$ and $v_{p}v_{r}\in E_{1}$. But then $i=j$ and since the ordering and the partition are strongly consistent in $G_{1}$, $v_{p}v_{q}\in E_{1}$ and so $(v_{p},w_{i})$ and $(v_{q},w_{j})$ are adjacent in $G_{1}\square G_{2}$. $\Box$

Corollary 20

If $G$ is (proper) $k$-thin then $G\square K_{t}$ is (proper) $kt$-thin. In particular, if $G$ is a (proper) interval graph then $G\square K_{t}$ is (proper) $t$-thin.

For a graph $G(V,E)$ and an integer $t$, we say that $f$ is a $t$-rainbow dominating function if it assigns to each vertex $v\in V$ a subset of $\{1,\dots,t\}$ such that $\cup_{u\in N(v)}f(u)=\{1,\dots,t\}$ for all $v$ with $f(v)=\emptyset$. Consider the following generalization of the dominating set problem.

$t$-rainbow domination problem

Instance: A graph $G=(V,E)$.

Find: a $t$-rainbow dominating function that minimizes $\sum_{v\in V}|f(v)|$.

The $t$-rainbow domination problem is equivalent to minimum dominating set of $G\square K_{t}$ [6]. As a consequence of Corollary 20 and the last remark in Section 4, it can be solved in polynomial time on graphs with bounded thinness for fixed values of $t$. This generalizes the polynomiality for interval graphs, recently proved by Hon, Kloks, Liu, and Wang in [20] (the algorithm for $t=2$ is claimed in [19]). The problem for proper interval graphs was stated as an open question by Brešar and Kraner Šumenjak in [6].

8 Conclusions and open problems

We described a wide family of combinatorial optimization problems that can be solved in polynomial time on classes of bounded thinness and bounded proper thinness. We think that some restrictions can be further generalized (specially the domination type ones), with more involved sets of parameters and transition rules. We tried to keep it as simpler as possible, yet including many of the classical combinatorial optimization problems in the literature.

We also proved a number of theoretical results, some of them related to the recognition problem for the classes, others relating the concept of thinness and proper thinness to other known graph parameters, and analyzing their behavior under the graph operations union, join, and Cartesian product.

Some open problems are the following.

Characterize (proper) $k$-thin graphs by minimal forbidden induced subgraphs (or at least within some graph class, we did it for thinness in cographs).

• 2.

Find sufficient conditions, for instance a family subgraphs to forbid as induced subgraphs, for a graph to be (proper) $k$-thin, even if these graphs are not necessarily forbidden induced subgraphs for (proper) $k$-thin graphs. These kind of results have been obtained for MIM-width in [23].

• 3.

Study the behavior of thinness under other graph products or graph operators in general.

• 4.

What is the complexity of computing the thinness/proper thinness of a graph? Or deciding if it is at most $k$ for some fixed values $k$?

• 5.

Can we develop some randomized algorithm to test just a subset of vertex orderings and obtain with high probability an approximation of the thinness/proper thinness?

• 6.

Can we improve the complexity of the algorithms that are in XP to FPT? Or prove a hardness result?

• 7.

Given a partition of the vertex set into a fixed number $k$ of classes, what is the complexity of deciding if there is a (strongly) consistent order for the vertices w.r.t. that partition (and finding it)? (We have proved that for an arbitrary number of classes the problems are NP-complete, and we have solved in polynomial time the symmetric problem, i.e., given the ordering, find a minimum (strongly) consistent partition.)

Acknowledgements: This work was partially supported by UBACyT Grant 20020130100808BA, CONICET PIP 112-201201-00450CO, ANPCyT PICT 2012-1324 and 2015-2218 (Argentina). We want to thank Gianpaolo Oriolo for bringing our attention into the thinness parameter and for helpful discussions about it, Daniela Saban for the example of an interval graph with high proper thinness, Nicolas Nisse for pointing out the relation between proper thinness and other interval graphs invariants to us, Boštjan Brešar and Martin Milanič for pointing out the open problem on rainbow domination to us, and Yuri Faenza for his valuable help in the improvement of the manuscript. Last but not least, we want to thank the anonymous referees for their useful comments that help us to improve the paper.