Largest reduced neighborhood clique cover number revisited

TL;DR

This paper revisits the properties of the largest reduced neighborhood clique cover number in graphs, exploring its bounds, connections to graph density, and conjectures about its behavior in incomparability graphs.

Contribution

It provides an overview of ${ ilde\beta}_t(G)$ properties, introduces bounded neighborhood clique cover graphs, and proposes conjectures on its exact value and separator theorems.

Findings

Connected ${\hat\beta}_t(G)$ to graph density measures.

Derived bounds for the neighborhood clique cover number.

Proposed conjectures on its value and separator properties.

Abstract

Let $G$ be a graph and $t\ge 0$. The largest reduced neighborhood clique cover number of $G$, denoted by ${\hat\beta}_t(G)$, is the largest, overall $t$-shallow minors $H$ of $G$, of the smallest number of cliques that can cover any closed neighborhood of a vertex in $H$. It is known that ${\hat\beta}_t(G)\le s_t$, where $G$ is an incomparability graph and $s_t$ is the number of leaves in a largest $t-$shallow minor which is isomorphic to an induced star on $s_t$ leaves. In this paper we give an overview of the properties of ${\hat\beta}_t(G)$ including the connections to the greatest reduced average density of $G$, or $\bigtriangledown_t(G)$, introduce the class of graphs with bounded neighborhood clique cover number, and derive a simple lower and an upper bound for this important graph parameter. We announce two conjectures, one for the value of ${\hat\beta}_t(G)$, and another for a separator theorem (with respect to a certain measure) for an interesting class of graphs, namely the class of incomparability graphs which we suspect to have a polynomial bounded neighborhood clique cover number, when the size of a largest induced star is bounded.

Full text

1 Introduction

This paper is a sequel to our paper [12]. We assume the reader is familiar with standard graph theory. Throughout this paper $G=(V,E)$ denotes an undirected graph. Recall that a graph $G$ is $k-$degenerate ($k\geq 0$), if every induced subgraph of $G$ has a vertex of degree at most $k$. Degeneracy of $G$ is the smallest integer $k$ so that $G$ is $k-$degenerate. Graphs with small degeneracy have nice structural and algorithm properties. Nesetril and Ossona de Mendez introduced an important graph parameter which is a generalization of degeneracy. In simple words they introduced the notion of the maximum edge density of a graph taken overall $t-$shallow minors.

A $t-$shallow minor, or a $t-$minor of $G$ in short, is a minor of $G$ which is obtained by contracting connected subgraphs of radius at most $t$, and deleting vertices (but not edges). Nesetril and Ossona de Mendez introduced the greatest reduced average density of $G$ (grad of $G$ in short), or $\bigtriangledown_{t}(G)$, to be the maximum edge density of any $t-$minor in $G$. It is easily seen that ${\hat{\delta}(G)\over 2}\leq\bigtriangledown_{t}(G)$, where $\hat{\delta}(G)$ is the degeneracy of $G$. They define $G$ to have bounded expansion, if $\bigtriangledown_{t}(G)$ is finite for every $t\geq 0$. They explored very nice structural and algorithmic properties of the class of bounded expansion graphs that contains many traditionally known “sparse” graphs, including the class of $H-$minor free graphs [9, 10, 8, 7].

We introduced the largest reduced neighborhood clique cover number of $G$, in [12]. Informally, consider the minimum number of disjoint cliques that covers the closed neighborhood of any vertex in a graph; Now take the maximum value of such a minimum overall $t-$minors of the graph. Formally, for a graph $H$, let $\beta(H)$ denote the clique cover number of $H$, that is, the minimum number of disjoint cliques that partitions $V(H)$. Now for any $x\in V(H)$, let $H_{x}$ denote the the closed neighborhood of $x$ in $H$, and note that $\beta(H_{x})\leq deg(x)$, where $deg_{H}(x)$ is the degree of $x$ in $H$. Now, let ${\tilde{\beta}(H)}=min_{x\in V(H)}\{\beta(H_{X})\}$, and note that ${\tilde{\beta}(H)}\leq{\hat{\delta}(H)}$. Next for any graph $G$ and $t\geq 0$ define largest reduced neighborhood clique cover number of $G$, denoted by ${\hat{\beta}}_{t}(G)$ to be the largest value of $\tilde{\beta}(H)$ for any $t-$minor $H$ of $G$. We say $G$ has a bounded neighborhood clique cover number if ${\hat{\beta}}_{t}(G)$ has a finite value for each $t\geq 0$. Note that $\hat{\beta}_{t}({K_{n}})={1}$ for any $t\geq 0$, nonetheless $\bigtriangledown_{t}(K_{n})={n-1\over 2}$. Furthermore, one can construct non trivial classes of graphs so that for every $G$ in the class ${\hat{\beta}}_{t}(G)$ is small, that is bounded by a constant, whereas, $\bigtriangledown_{t}(G)$ is arbitrary large. For instance, let $G=(V,E)$ be a connected graph which is the complement of a bipartite graph, where each partite class has $n$ vertices. Then ${\hat{\beta}}_{t}(G)\leq 2$, whereas, $\bigtriangledown_{t}(G)={|E|\over|V|}\geq{n-1\over 2}$, for any $t\geq 0$. Additionally, for any chordal graph $G$, ${\hat{\beta}}_{t}(G)=1$ [12], but of course one can construct very dense non trivial chordal graphs $G$ for which ${\hat{\beta}}_{t}(G)$ is unbounded.

${\hat{\beta}}_{t}(G)$, is an effective tool to study the properties of those graphs that are not “ sufficiently sparse”, to have a bounded expansion, but yet there is need to explore their properties. For instance, another interesting class of graphs for which $\hat{\beta}(G)$ is bounded, but grad of $G$ can be arbitrary large is the intersection graph of fact objects (spheres, cubes, boxes with bounded aspect ratio)[2] when geometric dimension is bounded. Specifically, see [12] for the following Theorem.

Theorem 1.1

Let $G$ be the intersection graph of fat objects in $R^{d}$ (spheres, cubes, boxes with bounded aspect ratio), then, ${\hat{\beta}}_{t}(G)=O(b^{d}.t^{2d})$, where $b$ is a constant that depends on the shape of the object. **

Section two contains a simple lower bound and an upper bound on ${\hat{\beta}}_{t}(G)$ in terms of the clique cover width of $G$, and some constructions that measures the ratio of the upper bound to the lower bound. Section three contains two conjectures related to incomparability graphs that arise from our studies here.

2 Bounds on ${\hat{\beta}}_{t}(G)$

It is interesting to observe that ${\hat{\beta}}_{0}(K_{n,n})=n$, therefore, ${\hat{\beta}}(K_{n,n})$ is not bounded. In fact, the following observation is easy to prove.

Observation 2.1

Let $p$ be the largest integer so that a $t-$shallow minor of $G$ is isomorphic to $K_{p,p}$, then ${\hat{\beta}}_{t}(G)\geq p$. **

For a clique cover $C$ in $G$, the clique cover graph of $C$ is the graph obtained by contracting the vertices of each clique in $C$ into a single vertex. The clique cover width of $G$, denoted by $CCW(G)$, is the minimum value of the bandwidth of all clique cover graphs in $G$[15, 13, 14]. In this paper when we write $C=\{C_{1},C_{2},....,C_{k}\}$, we mean $C$ is an ordered set. Let $ab$ be an edge width $a\in C_{i}$ and $b\in C_{j},j>i$, and let $W(e)=j-i$. We call $W(e)$ the width of $e$. An important application of the clique cover width is in the derivation of separation theorems in dense graphs, where separation can be defined for other types measures [15], instead of just the number of vertices. Recall that according to the planar separation theorem, any $n$ vertex planar graph can be separated into two subgraphs, each having at most $2n/3$ vertices, by removing $O({\sqrt{n}})$ vertices. Any $G$ can be separated with respect to an optimal (or feasible) set $C$of cliques (utilizing $CCW(G)$): There is partition of $\{A,S,B\}$ of $V(G)$ so that $(i)$ there are no edges between $A$ and $B$, $(ii)$ $S$ can covered with at most $CCW(G)$ many cliques from $C$, and $(iii)$ $A$ and $B$ are each covered with at most $2|C|/3$ cliques from $C$[15, 14].

Theorem 2.1

For any graph $G$, ${\hat{\beta}}_{t}(G)\leq k+1$, where $k$ is the largest clique cover width of any $t-$shallow minor of $G$.

Proof. Let $\{C_{1},C_{2},....,C_{K}\}$ be a clique cover of a graph $H$. Let $e_{a}=ab,a\in C_{1},b\in C_{i}$ be an edge of largest width incident to $a$. Let $e^{*}$ be an edge having an end point in $C_{1}$ with $W(e^{*})=\min\{W(e_{a})|a\in C_{1}\}$. By definition of $e^{*}$, $H_{a}$ can be covered with $W(e^{)}*+1$ cliques, and hence ${\tilde{\beta}(H)}\leq W(e^{*})+1$. Therefore ${\tilde{\beta}(H)}\leq CCW(H)+1$, since $CCW(H)\geq W(e^{*})$. To finish the proof take $H$ to be a $t$-minor of $G$. $\Box$.

Corollary 2.1

Let $k$ denote the largest clique cover width of any $t-$shallow minor of $G$, and $p$ be largest integer so that any $t-$shallow minor of $G$ is isomorphic to $K_{p,p}$. Then, $p\leq{\hat{\beta}}_{t}(G)\leq k+1$.

It is easy to verify that $CCW(H)\leq CCW(G)$, for any induced subgraph $H$ of $G$. Nonetheless, for a $t-$minor $H$ of $G$, $CCW(H)$, or $k$ in Corollary, 2.1 may be much larger than $CCW(G)$. Generally speaking, it would nice to know how large the ratio $k/p$ may be.

Observation 2.2

For any $t\geq 0$, and $n>t$, there is an $n$ vertex graph $G$, with $CCW(G)=1$, so that for a $t-$minor $H$ of $G$, $t\geq CCW(H)\geq t/2$. Moreover, in this case, neither $G$, nor $H$ contain $K_{2,2}$ as an induced subgraph.

Justification. Let $P_{n}$ be a path of $n$ vertices on vertex set $X=\{x_{1},x_{2},...,x_{n}\}$. Now let $S$ be a an independent set of $n$ vertices. To construct $G=(V,E)$ place a perfect matching of cardinality $n$ between $S$ and $X$. It is easily verified that $CCW(G)=1$. Now for a given $n\geq t\geq 0$, contract $x_{1},x_{2},...,x_{t}$ into one single vertex to obtain a $t-$minor $H$. Observe that $H$ has an induced star on $t$ vertices. Thus, $CCW(H)\geq t/2$ [13]. Furthermore, it is not difficult to see that $G$ is an incomparability graph (a graph whose complement has a transitive orientation on edges), and so is $H$, since $H$ is obtained by contractions of edges in $G$. Since $H$ is an incomparability graph we must have $CCW(H)\leq s$, where $s$ is the number of leaves in a largest induced star [13]. Finally, it is easy to verify that neither $H$ or $G$ have $K_{2,2}$ as a subgraph, since $G$ is acyclic. $\Box$

Observation 2.3

For any $t\geq 0$, and $n>>t$, there is a graph $G$, on $n+t(t+1)$ vertices that excludes $K_{2,2}$ as an induced subgraph, but has a $t-$minor $H$ that contains $K_{t+1,t+1}$ as an induced subgraph. Moreover, $CCW(G)\geq n/2$.**

Justification. Let $V(G)=A\cup_{i=1}^{t+1}B_{i}$, where $A$ is a independent set of size $t+1$, and for $i=1,2,...,t$ each $B_{i}$ is path on $t+1$ vertices; $B_{t+1}$ is a cycle on $n$ vertices. Now for each $i=1,2,...,t+1$ add a perfect matching of size $t$ between vertices in $A$ and vertices in $B_{i}$. Thus each vertex in $A$ has degree $t$, where for $i=1,2,...,t$, each vertex of $B_{i}$ has degree at most $3$. Note that $G$ does not have $K_{2,2}$ as an induced subgraph. Furthermore, since $B_{t+1}$ is a cycle of $n$ vertices, we have $CCW(G)\geq n/2$. Now for $i=1,2,...,t$, contract each path $B_{i}$ into a single vertex. For $B_{i+1}$ contract the first $t+1$ vertices to a vertex. Then the resulting graph $H$ has an induced subgraph isomorphic to $K_{t+1,t+1}$. $\Box$

3 Incomparability graphs

Recall that a chordal graph does not have any chord-less cycles [3]. An incomparability graph is a graph whose complement has a transitive orientation [16]. Incomparability graphs are perfect, have geometric realizations, and have recently been subject to intense investigations, due to their intimate connections to string graphs. One wonders if there is a meaningful converse to Observation 2.1. That is, can one find a suitable upper bound on ${\hat{\beta}}_{t}(G)$ that is related to the lower bound in 2.1? It is less likely that this is the case for all graphs, nonetheless, we suspect that there is a weak converse to 2.1 when $G$ is an incomparability graph. Specifically, we have shown that if an incomparability graph $G$ does not have a $t-$shallow minor which is isomorphic to an induced star on $s_{t}$ leaves, then, ${\hat{\beta}}_{t}(G)\leq s_{t}$. Moreover, we have shown that for any incomparability graph $G$, ${s\over 2}\leq CCW(G)\leq s$, where $s$ is the number of leaves in a largest induced star in $G$. Hence, a natural question is how large $s_{t}/s$ can be?

Conjecture 3.1

Let $G$ be an incomparability graph that does not have an induced star which is isomorphic to an induced star on $s$ leaves. Then, the size of a largest induced star $s_{t}$ in any $t-$shallow minor of $G$ is at most $O(t.s)$. Consequently, ${\hat{\beta}}_{t}(G)=O(t.s)$, for any $t\geq 0$.

If the above conjecture were to be true, then ${\hat{\beta}}_{t}(G)=O(t.s)$, where $t$ is the number of leaves in a largest induced star in $G$. Note that the conjecture implies that the class of incomparability graphs have a linearly bounded neighborhood clique cover number, when the size of a largest induced star is fixed.

By observation 2.1, ${\hat{\beta}}_{t}(G)\geq p_{t}$, where $p_{t}$ is the largest integer so that $K_{p_{t},p_{t}}$ is a $t-$shallow minor of $G$. Hence o get a good estimate for ${\hat{\beta}}_{t}(G)$ (if the conjecture were to be true), one has to investigate how large $t.s/p_{t}$ can be.

It is easy to observe that If $G$ is a chordal graph, then, ${\hat{\beta}}_{t}(G)=1$ [12]. Moreover, the separation property with respect to cliques holds for any chordal graph $G$, regardless of the value the clique cover width. Particularly, given a clique tree [3] of $G$ associated with a set $C$ of maximal cliques, there is one clique $B$ in $C$, so that after removal of $B$, each the two remaining (separated) subgraph of $G$ can be covered by at most ${2|C|/3}$ cliques from $C$. Now let $G$ be an interval graph; Since $G$ is chordal ${\hat{\beta}}_{t}(G)=1$, and additionally $G$ has the stated separation property. Particularly, note that $G$ is chordal and also an incomparability graph that does not have a $K_{2,2}$ as an induced subgraph. In fact, no $t-$minor of an interval graph $G$ can have $K_{2,2}$ as an induced subgraph. So one can suspect that if a incomparability graph $G$ does not have a large $K_{p,p}$ as a $t-$minor, then, $G$ has nice separation properties with respect to cliques.

Conjecture 3.2

Let $p$ be fixed, and let $G$ be an incomparability graph that does not have $K_{p,p}$ as a $t-$shallow minor. Then, there is a clique cover $C$ in $G$ so that the removal of $O({\sqrt{|C|}})$ cliques from $C$, separates $G$ into two subgraphs so that each subgraph can be covered with at most ${2|C|/3}$ cliques from $C$.

We remark that by a general result of Fox and Pach [4] (see also an earlier result of Bodlaender and Thilikos on $k-$chordal graphs [1]), any incomparability graph $G$ on $n$ vertices and $m$ edges has a separation $(L,S,R)$ so that $S=O({\sqrt{m}})$, and $|L|,|S|\leq 2n/3$, but conjecture 3.2 does not follow from these result.