Reconfiguration Graph for Vertex Colourings of Weakly Chordal Graphs

TL;DR

This paper investigates the structure and properties of reconfiguration graphs for vertex colourings in weakly chordal graphs, revealing disconnectedness in certain cases and polynomial diameter bounds in a new subclass called compact graphs.

Contribution

It introduces the class of $k$-colourable compact graphs and analyzes their reconfiguration graph diameters, extending understanding of colourings in weakly chordal graphs.

Findings

Existence of $k$-colourable weakly chordal graphs with disconnected reconfiguration graphs for $k+1$ colours.

Polynomial diameter bound ($O(n^2)$) for reconfiguration graphs of $k$-colourable compact graphs.

Contains all $k$-colourable co-chordal graphs and specific $(P_5, ar{P}_5, C_5)$-free graphs for $k=3$.

Abstract

The reconfiguration graph $R_k(G)$ of the $k$-colourings of a graph $G$ contains as its vertex set the $k$-colourings of $G$ and two colourings are joined by an edge if they differ in colour on just one vertex of $G$. We show that for each $k \geq 3$ there is a $k$-colourable weakly chordal graph $G$ such that $R_{k+1}(G)$ is disconnected. We also introduce a subclass of $k$-colourable weakly chordal graphs which we call $k$-colourable compact graphs and show that for each $k$-colourable compact graph $G$ on $n$ vertices, $R_{k+1}(G)$ has diameter $O(n^2)$. We show that this class contains all $k$-colourable co-chordal graphs and when $k = 3$ all $3$-colourable $(P_5, \overline{P_5}, C_5)$-free graphs. We also mention some open problems.

Figures

Full text

We show that for each $k\geq 3$ there is a $k$-colourable weakly chordal graph $G$ such that $R_{k+1}(G)$ is disconnected. We also introduce a subclass of $k$-colourable weakly chordal graphs which we call $k$-colourable compact graphs and show that for each $k$-colourable compact graph $G$ on $n$ vertices, $R_{k+1}(G)$ has diameter $O(n^{2})$. We show that this class contains all $k$-colourable co-chordal graphs and when $k=3$ all $3$-colourable $(P_{5},\overline{P_{5}},C_{5})$-free graphs. We also mention some open problems.

1 Institutt for informatikk,

Universitetet i Bergen, Norway

2 Department of Applied Mathematics,

Charles University, Prague

1 Introduction

Let $G$ be a graph, and let $k$ be a non-negative integer. A $k$-colouring of $G$ is a function $f:V(G)\rightarrow\{1,\dots,k\}$ such that $f(u)\not=f(v)$ whenever $(u,v)\in E(G)$. The reconfiguration graph $R_{k}(G)$ of the $k$-colourings of $G$ has as vertex set the set of all $k$-colourings of $G$ and two vertices of $R_{k}(G)$ are adjacent if they differ on the colour of exactly one vertex (the change of the colour is the so called colour switch). For a positive integer $\ell$, the $\ell$-colour diameter of a graph $G$ is the diameter of $R_{\ell}(G)$.

In the area of reconfigurations for colourings of graphs, one focus is to determine the complexity of deciding whether two given colourings of a graph can be transformed into one another by a sequence of recolourings (that is, to decide whether there is a path between these two colourings in the reconfiguration graph); see, for example, [9, 8, 6, 4]. Another focus is to determine the diameter of the reconfiguration graph in case it is connected or the diameter of its components if it is disconnected [3, 7, 2, 5, 11]. We refer the reader to [15, 13] for excellent surveys on reconfiguration problems.

In this note, we continue the latter line of study of reconfiguration problems. In Section 3, we shall show that the $(k+1)$-colour diameter of $k$-colourable weakly chordal graphs can be infinite. On the positive side, in Section 4, we shall consider two specific subclasses of $k$-colourable perfect graphs and show that their $(k+1)$-colour diameter is quadratic in the order of the graph.

2 Preliminaries

For a graph $G=(V,E)$ and a vertex $u\in V$, let $N_{G}(u)=\{v:uv\in E\}$. A separator of a graph $G=(V,E)$ is a set $S\subset V$ such that $G-S$ has more connected components than $G$. If two vertices $u$ and $v$ that belong to the same connected component in $G$ are in two different connected components of $G-S$, then we say that $S$ separates $u$ and $v$. A chordless path $P_{n}$ of length $n-1$ is the graph with vertices $v_{1},\dots,v_{n}$ and edges $v_{i}v_{i+1}$ for $i=1,\dots,n-1$. It is a cycle $C_{n}$ of length $n$ if the edge $v_{1}v_{n}$ is also present.

The complement of $G$ is denoted $\overline{G}=(V,\overline{E})$. It is the graph on the same vertex set as $G$ and there is an edge in $G$ between two vertices $u$ and $v$ if and only if there is no edge between $u$ and $v$ in $\overline{G}$. A set of vertices in a graph is anticonnected if it induces a graph whose complement is connected. A clique or a complete graph is a graph where every pair of vertices is joined by an edge. The size of a largest clique in a graph $G$ is denoted $\omega(G)$. The chromatic number $\chi(G)$ of a graph $G$ is the least integer $k$ such that $G$ is $k$-colourable.

A graph $G$ is perfect if $\omega(G^{\prime})=\chi(G^{\prime})$ for every (not necessarily proper) subgraph $G^{\prime}$ of $G$. A hole in a graph is a cycle of length at least 5 and an antihole is the complement of a hole. A graph is perfect if it is (odd hole, odd antihole)-free [10]. A graph is weakly chordal if it is (hole, antihole)-free. A graph is co-chordal if it is ($\overline{C_{4}}$, anti-hole)-free. Every weakly chordal graph is perfect. Every co-chordal graph and every $(P_{5},\overline{P_{5}},C_{5})$-free graph is weakly chordal.

A 2-pair of a graph $G$ is a pair $\{x,y\}$ of nonadjacent distinct vertices of $G$ such that every chordless path from $x$ to $y$ has length 2. We often use the following well-known lemma:

Lemma 2.1** (Hayward et al. [12]).**

A graph $G$ is weakly chordal graph if and only if every subgraph of $G$ is either a complete graph or it contains a 2-pair.

3 Weakly chordal graphs

In this section, we establish the following result.

Theorem 3.1**.**

For each $k\geq 3$ there exists a $k$-colourable weakly chordal graph $G_{k}$ such that $R_{k+1}(G_{k})$ is disconnected.

The graph $G_{k}$ is depicted in Figure 1.

In other words, Theorem 3.1 states that for each $k\geq 3$ the $(k+1)$-colour diameter of $k$-colourable weakly chordal graphs can be infinite. It is worth mentioning that the case $k=2$ is already known [3] as the class of 2-colourable weakly chordal graphs is precisely the class of chordal bipartite graphs. It is also worth mentioning that Bonamy, Johnson, Lignos, Patel and Paulusma [3] asked whether the $(k+1)$-colour diameter of $k$-colourable perfect graphs is connected. This was answered negatively in [1] – the counterexample consists of a complete bipartite graph minus a matching. Our Theorem 3.1 thus strengthens this counterexample.

Proof of Theorem 3.1.

It suffices to construct for each $k\geq 3$ a $k$-colourable weakly chordal graph $G_{k}$ and a $(k+1)$-colouring of $G$ such that each of the $k+1$ colours appear in the closed neighbourhood of every vertex of $G_{k}$, as then no vertex of $G_{k}$ can get recoloured.

Such graph is depicted in Figure 1. It is formed from the disjoint union of four complete graphs $K_{k-1}$, one on vertices $u_{i}$ for $i\in\{1,\dots,k-1\}$, the other three on vertices $v_{i},w_{i},z_{i}$, respectively, and two further vertices $x$ and $y$. These parts are joined together by additional edges such that $x$ and $y$ are connected to each $u_{i}$ and to each $v_{i}$; $u_{1}$ and $v_{1}$ are connected to each $z_{i}$ and to each $w_{i}$; and finally, $G_{k}$ contains two further edges $xz_{1}$ and $yw_{1}$.

A possible $k$-colouring of $G_{k}$ is schematically shown on the left side of Figure 1; in both pictures each 4-tuple $u_{i},v_{i},w_{i},z_{i}$ for $i\in\{2,\dots,k-2\}$ receives a unique colour. On the other hand, in the $(k+1)$-colouring depicted on the right, every vertex of $G_{k}$ has its neighbours coloured by the $k$ remaining colours, hence no vertex can be recoloured. Hence, this colouring corresponds to an isolated vertex in the reconfiguration graph, and thus the reconfiguration graph $R_{k+1}(G_{k})$ is disconnected.

It remains to show that $G_{k}$ is weakly chordal. Observe first that any distinct $u_{i},u_{j}$ with $i,j\in\{2,\dots,k-1\}$ have the same neighbourhood, hence no hole and no antihole may contain both of them. The same holds for vertices $v_{i},w_{i},z_{i}$, respectively. Hence without loss of generality we may assume that no vertex with index at least three participates in a hole nor in an antihole. In other words, it suffices to restrict ourselves only to the graph $G_{3}$ to show that it is weakly chordal.

By examining possible paths, one can also realise that vertices $x,y$ as well as $u_{1},v_{1}$ form a 2-pair. Since no hole may contain a 2-pair, we may assume without loss of generality that a possible hole does not contain $y$ and $v_{1}$. The vertex $x$ separates $v_{2}$ and also the vertex $u_{1}$ separates $w_{1},w_{2}$ from the rest in the graph $G_{3}-\{y,v_{1}\}$, hence $v_{2},w_{1},w_{2}$ also do not belong to a hole. We were left with five vertices $x,u_{1},u_{2},z_{1},z_{2}$ which do not induce a hole, hence $G_{3}$ does not contain a hole at all.

Now assume for a contradiction that $G_{3}$ contains an antihole on at least 6 vertices. The graph $G_{3}$ contains only 6 vertices of degree at least 4, hence the antihole contains exactly 6 vertices. The neighbourhood of $u_{2}$ induces a diamond (a $K_{4}$ minus an edge), hence it does not belong to the antihole as no diamond is an induced subgraph of $\overline{C_{6}}$. The same holds for $v_{2},z_{2},w_{2}$. We are left with vertices $x,y,u_{1},v_{1},w_{1},z_{1}$ which do not induce an antihole in $G_{3}$, hence $G_{3}$ does not contain an antihole at all. Since $G_{3}$ is weakly chordal, we have shown that $G_{k}$ is also weakly chordal for each $k\geq 4$. ∎

4 Quadratic diameter

In this section, we introduce a subclass of $k$-colourable weakly chordal graphs that we call $k$-colourable compact graphs. We show in Theorem 4.1 that for each $k$-colourable compact graph $G$ on $n$ vertices the diameter of $R_{k+1}(G)$ is $O(n^{2})$. We then show in Lemma 4.1 that $k$-colourable co-chordal graphs are $k$-colourable compact and in Lemma 4.2 that $3$-colourable $(P_{5},\overline{P_{5}},C_{5})$-free graphs are $3$-colourable compact.

For a 2-pair $\{u,v\}$ of a weakly chordal graph $G$, let $S(u,v)=N_{G}(u)\cap N_{G}(v)$. Note that, by the definition of a 2-pair, $S(u,v)$ is a separator of $G$ that separates $u$ and $v$. Let $C_{v}$ denote the component of $G\setminus S(u,v)$ that contains the vertex $v$.

Definition 4.1**.**

A weakly chordal graph $G$ is said to be compact if every subgraph $H$ of $G$ either

• (i)

is a complete graph, or

• (ii)

contains a 2-pair $\{x,y\}$ such that $N_{H}(x)\subseteq N_{H}(y)$, or

• (iii)

contains a 2-pair $\{x,y\}$ such that $C_{x}\cup S(x,y)$ induces a clique on at most three vertices.

Theorem 4.1**.**

Let $k$ be a positive integer, and let $G$ be a $k$-colourable compact graph on $n$ vertices. Then $R_{k+1}(G)$ has diameter $O(n^{2})$.

Proof.

It suffices to show that we can recolour a $(k+1)$-colouring $\alpha$ of $G$ to a $(k+1)$-colouring $\beta$ by recolouring every vertex at most $2n$ times.

We first suppose that $G$ is a complete graph. In this case, we know from [3] that we can recolour $\alpha$ to $\beta$ by recolouring every vertex at most $2n$ times. We now consider the case when $G$ is not a complete graph but satisfies condition (ii) of compact graphs. We use induction on the number of vertices of $G$. Let $\{x,y\}$ be a 2-pair of $G$ such that $N_{G}(x)\subseteq N_{G}(y)$. From $\alpha$ and $\beta$, we can immediately recolour $x$ with, respectively, $\alpha(y)$ and $\beta(y)$. Let $G^{\prime}=G-\{x\}$, and let $\alpha_{G^{\prime}}$ and $\beta_{G^{\prime}}$ denote the restrictions of $\alpha$ and $\beta$ to $G^{\prime}$. By our induction hypothesis, we can transform $\alpha_{G^{\prime}}$ to $\beta_{G^{\prime}}$ by recolouring every vertex at most $2(n-1)$ times. We can extend this sequence of recolourings to a sequence of recolourings in $G$ by recolouring $x$ using the same recolouring as $y$. Then $x$ gets recoloured as many times as $y$ as needed.

Suppose that $G$ satisfies condition (iii) of compact graphs. We use induction on the number of vertices. If $S(x,y)$ contains exactly two vertices, then $C_{x}=\{x\}$ and hence $G$ satisfies condition (ii) of compact graphs. So we can assume that $S(x,y)$ is a single vertex $z$ and $C_{x}$ consists of $x$ and another vertex $w$, see Figure 2. From $\alpha$ and $\beta$, we can recolour $x$ and $w$ to another colour either immediately or by first recolouring $w$ and $x$, respectively. Let $G^{*}=G-\{x,w\}$. By our induction hypothesis, we can transform $\alpha_{G^{*}}$ to $\beta_{G^{*}}$ by recolouring every vertex at most $2(n-2)$ times. We can extend this sequence of recolourings to a sequence of recolourings in $G$ by recolouring $x$ and $w$ whenever $z$ is recoloured to their colour. At the end of the sequence we recolour $x$ and $w$ so that they agree in both colourings. As $x$ and $w$ are recoloured at most two more times as $z$, this completes the proof. ∎

Lemma 4.1**.**

Every $k$-colourable co-chordal graph is compact.

Proof.

Let $G$ be a $k$-colourable co-chordal graph. If $G$ is a complete graph, then $G$ is compact by definition. Otherwise, since $G$ is weakly chordal, $G$ contains a 2-pair $\{x,y\}$ by Lemma 2.1. If $x$ has a neighbour $x_{1}$ that is not a neighbour of $y$ and $y$ has a neighbour $y_{1}$ that is not a neighbour of $x$, then $x_{1}$ is not adjacent to $y_{1}$, as otherwise $S(x,y)$ does not separate $x$ and $y$. But then the edges $xx_{1}$ and $yy_{1}$ form $\overline{C_{4}}$, a contradiction. Therefore, $N_{G}(x)\subseteq N_{G}(y)$ or vice-versa and hence the graph $G$ is compact as required. ∎

Lemma 4.2**.**

Every $3$-colourable $(P_{5},\overline{P_{5}},C_{5})$-free graph is compact.

The proof of this lemma will require a little more work. First, we need some definitions and auxiliary results. When $T$ is a set of vertices of a graph $G$, a set $D\subseteq V(G)\setminus T$ is $T$-complete if each vertex of $D$ is adjacent to each vertex of $T$. Let $D(T)$ denote the set of all $T$-complete vertices.

Lemma 4.3** (Trotignon and Vušković [14]).**

Let $G$ be a weakly chordal graph, and let $T\subseteq V(G)$ be a set of vertices such that $G[T]$ is anticonnected and $D(T)$ contains at least two non-adjacent vertices. If $T$ is inclusion-wise maximal with respect to these properties, then any chordless path of $G\setminus T$ whose ends are in $D(T)$ has all its vertices in $D(T)$.

The following corollary is implicit in [14].

Corollary 4.1**.**

Let $G$ be a weakly chordal graph that contains a chordless path $P$ of length $2$. Then there exists an anticonnected set $T$ containing the centre of $P$, such that $D(T)$ contains a 2-pair of $G$.

In particular, the 2-pair can always be found in the neighbourhood of the centre of $P$.

Proof.

We start with the centre of $P$ to build our set $T$ as in Lemma 4.3. Then $D(T)$ is not a clique as it contains both ends of $P$. Hence, by definition of weakly chordal graphs, $D(T)$ contains a 2-pair. This 2-pair is also a 2-pair of $G$ by Lemma 4.3. ∎

We are now ready to prove Lemma 4.2.

Proof of Lemma 4.2.

Let $G$ be a 3-colourable $(P_{5},\overline{P_{5}},C_{5})$-free graph. Suppose towards a contradiction that $G$ is not compact. In particular, $G$ is not a complete graph, as otherwise $G$ would be compact by the definition. Since $G$ is weakly chordal, it contains a 2-pair $\{x,y\}$. We choose $\{x,y\}$ such that $|V(C_{x})|$ is minimum over all 2-pairs $\{x,y\}$ of $G$.

Denote by $G^{\prime}$ the subgraph of $G$ induced by the union of $S(x,y)$ and the vertices of $C_{x}$. The subgraph $G^{\prime}$ is not complete, as $G$ would be compact, hence $G^{\prime}$ contains a chordless path of length 2. Let us argue that $G^{\prime}$ contains a chordless path of length 2 whose centre is, in fact, a member of $C_{x}$.

If $C_{x}$ is not complete, then this is immediate. And if $S(x,y)$ is not complete, then it contains a pair of vertices $u$ and $v$ that are not adjacent, so we take $u$, $x$, $v$ to be our path. Hence we can assume that $C_{x}$ and $S(x,y)$ are both complete and, as $G^{\prime}$ is not complete, there must be a vertex $u$ of $C_{x}$ and a vertex $v$ of $S(x,y)$ such that $uv\not\in E(G)$. Then we can take $u$, $x$, $v$ as our path and our aim is achieved.

Applying Corollary 4.1 with $P$ being a chordless path of length 2 whose centre is in $C_{x}$, we find that $G^{\prime}$ contains a 2-pair $\{z,w\}$ that is also a 2-pair of $G$.

We next want to argue that $z,w\in S(x,y)$. For a contradiction, assume without loss of generality that $z$ belongs to $C_{x}$. This implies that $S(z,w)\subseteq V(G^{\prime})$. So there must be a component $C_{1}$ of $G\setminus S(z,w)$ such that $C_{1}$ and $S(x,y)$ do not have a vertex in common since $y$ is adjacent to every vertex of $S(x,y)$. Therefore, we find that $C_{1}\subseteq C_{x}$. If $C_{1}=C_{x}$, then $S(x,y)=S(z,w)$ and hence $z,w\in C_{1}$ which is impossible because $S(z,w)$ separates $z$ and $w$. Therefore, $|V(C_{1})|<|V(C_{x})|$ holds, which contradicts our choice of $C_{x}$.

Hence we have concluded that $z,w\in S(x,y)$. Now the vertices $x,z,y,w$ form a cycle such that $\{x,y\}$ and $\{z,w\}$ are 2-pairs. Since $G$ is not compact, $N_{G}(x)\not\subseteq N_{G}(y)$, hence there exists a vertex $x^{\prime}$ that is adjacent to $x$ but not to $y$. Analogously, there are vertices $y^{\prime},z^{\prime},w^{\prime}$ such that $yy^{\prime},zz^{\prime},ww^{\prime}\in E$, but $xy^{\prime},wz^{\prime},zw^{\prime}\notin E$. If $x^{\prime}=z^{\prime}$, then $x^{\prime}$ must be adjacent to $y$ or $w$ else $x^{\prime},x,z,y,w$ form $\overline{P_{5}}$. But $x^{\prime}$ is not adjacent to $y$ and $z^{\prime}$ not adjacent to $w$, thus $x^{\prime}\not=z^{\prime}$. Similarly, $y^{\prime}\not=w^{\prime}$ and hence $x^{\prime},y^{\prime},w^{\prime},z^{\prime}$ are distinct. Moreover, since $S(x,y)$ is a separator, we get $x^{\prime}y^{\prime}\notin E$ and analogously $w^{\prime}z^{\prime}\notin E$, see Figure 3 a).

If both $z^{\prime}$ and $w^{\prime}$ are not adjacent to $x$, then $z^{\prime},z,x,w,w^{\prime}$ form $P_{5}$. So we can assume without loss of generality that $z^{\prime}$ is adjacent to $x$. If $z^{\prime}$ is not adjacent to $y$, then $z^{\prime},z,x,w,y$ would form $\overline{P_{5}}$ as $w$ is not adjacent to $z^{\prime}$. Thus $z^{\prime}$ is adjacent to both $x$ and $y$.

Similarly, to avoid $P_{5}$ on vertices $x^{\prime},x,z,y,y^{\prime}$ either $x^{\prime}$ or $y^{\prime}$ must be adjacent to $z$ and hence also to $w$, so we assume without loss of generality that $x^{\prime}$ is adjacent to $z$ and $w$. If $x^{\prime}$ is adjacent to $z^{\prime}$, then $x,z,z^{\prime},x^{\prime}$ form $K_{4}$, a contradiction with the assumption the $G$ is 3-colourable, see Figure 3 b).

To avoid a $P_{5}$ on the vertices $x^{\prime},x,z^{\prime},y,y^{\prime}$, the vertices $z^{\prime}$ and $y^{\prime}$ are forced to be adjacent. Similarly $x^{\prime}w^{\prime}\in E$ as otherwise $z^{\prime},z,x^{\prime},w,w^{\prime}$ induce a $P_{5}$. To avoid a $K_{4}$ on the vertices $z,z^{\prime},y,y^{\prime}$, the vertices $z$ and $y^{\prime}$ need to be nonadjacent. Similarly $xw^{\prime}\notin E$ as otherwise $x,x^{\prime},w,w^{\prime}$ induce a $K_{4}$.

Now the vertices $w^{\prime},x^{\prime},x,z^{\prime},y^{\prime}$ induce either a $C_{5}$ or a $P_{5}$, depending whether the edge $y^{\prime}w^{\prime}$ is present or not, see Figure 3 c). In either case we arrive at a contradiction and the lemma is proved. ∎

We are aware the concept of compact graphs does not fit tight with the class of $(P_{5},\overline{P_{5}},C_{5})$-free graphs, as some of these graphs need not to be $k$-colourable compact graphs for $k\geq 4$. An example of such graph $H$ for $k=4$ is depicted in Figure. 4.

Due to symmetries of the graph $H$ it suffices without loss of generality to consider only the 2-pair $\{x,y\}$ as other 2-pairs could be mapped onto $\{x,y\}$ by an automorphism of $H$. Observe that this 2-pair violates the conditions of the definition 4.1 for $H$ to be 4-colourable compact.

Any choice of five vertices from $H$ would contain two vertices joined by a horizontal or a vertical edge, and such edge cannot be extended to an induced $P_{3}$, hence $H$ is also $P_{5}$-free. Also, such choice of five vertices would contain two opposite vertices either of the inner $C_{4}$ or from the outer one, like the vertices $x$ and $y$. As such two vertices form an $2$-pair, $H$ contains no $C_{5}$. Finally, $H$ has only two induced $C_{4}$ and neither could be completed by any fifth vertex to a $\overline{P_{5}}$.

5 Concluding remarks

We end this note with two open problems.

Problem 1**.**

For which integer $\ell>k+1$ is the $\ell$-colour diameter of $k$-colourable weakly chordal graphs connected?

Problem 2**.**

Is it true that the $(k+1)$-colour diameter of $k$-colourable $(P_{5},\overline{P_{5}},C_{5})$-free graphs is quadratic for each $k\geq 4$?