List k-Colouring P_t-Free Graphs with No Induced 1-Subdivision of K_1,s: a Mim-width Perspective

1 Introduction

Width parameters play an important role in algorithmic graph theory, as evidenced by various surveys [12, 18, 19, 30, 31]. A graph class $G$ has bounded width, for some width parameter, if there exists a constant $c$ such that every graph in $G$ has width at most $c$ . Mim-width is a relatively young width parameter that was introduced by Vatshelle [34]. It is defined as follows. A branch decomposition for a graph $G$ is a pair $(T, δ)$ , where $T$ is a subcubic tree and $δ$ is a bijection from $V (G)$ to the leaves of $T$ . Every edge $e \in E (T)$ partitions the leaves of $T$ into two classes, $L_{e}$ and $_{e}$ , depending on which component of $T - e$ they belong to. Hence, $e$ induces a partition $(A_{e},_{e})$ of $V (G)$ , where $δ (A_{e}) = L_{e}$ and $δ (_{e}) =_{e}$ . We let $G [A_{e},_{e}]$ be the bipartite subgraph of $G$ induced by the edges with one end-vertex in $A_{e}$ and the other in $_{e}$ . A matching $F \subseteq E (G)$ of $G$ is induced if there is no edge in $G$ between vertices of different edges of $F$ . We let ${c u t m i m}_{G} (A_{e},_{e})$ be the size of a maximum induced matching in $G [A_{e},_{e}]$ . The mim-width ${m i m w}_{G} (T, δ)$ of $(T, δ)$ is the maximum value of ${c u t m i m}_{G} (A_{e},_{e})$ over all edges $e \in E (T)$ . The mim-width $m i m w (G)$ of $G$ is the minimum value of ${m i m w}_{G} (T, δ)$ over all branch decompositions $(T, δ)$ for $G$ .

Vatshelle [34] proved that every class of bounded clique-width, or equivalently, bounded boolean-width, module-width, NLC-width or rank-width, has bounded mim-width, and that the converse is not true. That is, he proved that there exist graph classes of bounded mim-width that have unbounded clique-width. This means that proving that a problem is polynomial-time solvable for graph classes of bounded mim-width (see [1, 2, 3, 7, 13, 21, 22, 23] for examples of such problems) yields more tractable graph classes than doing this for clique-width. Hence, mim-width has greater modeling power than clique-width. However, the trade-off is that fewer problems admit such an algorithm, as we explain below by means of a relevant example, namely the classical Colouring problem. Moreover, computing mim-width is NP-hard [33] and, in contrast to the situation of clique-width [32], it remains a challenging open problem to develop a polynomial-time algorithm for approximating the mim-width of a graph (doing this within a constant factor of the optimal is not possible unless $N P = Z P P$ [33]). This has the following implication. For a class of graphs $G$ with bounded mim-width, we need an explicit polynomial-time algorithm for computing a branch decomposition whose mim-width is bounded by a constant. When this is possible, we say that the mim-width of $G$ is quickly computable. We can then develop a polynomial-time algorithm for the problem of interest via dynamic programming over the computed branch decomposition; see [5, 7, 24] for some examples.

A colouring of a graph $G = (V, E)$ is a mapping $c : V \to {1, 2, \dots}$ that gives each vertex $u \in V$ a colour $c (u)$ in such a way that, for every two adjacent vertices $u$ and $v$ , we have that $c (u) \neq c (v)$ . If for every $u \in V$ we have $c (u) \in {1, \dots, k}$ , then we say that $c$ is a $k$ -colouring of $G$ . The Colouring problem is to decide whether a given graph $G$ has a $k$ -colouring for some given integer $k \geq 1$ . If $k$ is fixed, that is, not part of the input, we call this the $k$ -Colouring problem. A classical result of Lovász [29] states that $k$ -Colouring is NP-complete even if $k = 3$ .

The Colouring problem is an example of a problem that distinguishes between classes of bounded mim-width and bounded clique-width: it is polynomial-time solvable for every graph class of bounded clique-width [26] but NP-complete for circular-arc graphs [14], which form a class whose mim-width is bounded and quickly computable [1]. When we fix $k$ , we no longer have this distinction, as $k$ -Colouring, for every fixed integer $k \geq 1$ , is polynomial-time solvable for a graph class whose mim-width is bounded and quickly computable [7]. Kwon [28] observed that an instance of the more general List $k$ -Colouring problem (defined below) can be reduced to an instance of $k$ -Colouring while only increasing the mim-width of the input graph by at most $k$ (see also [6]). Hence, the List $k$ -Colouring problem is also polynomial-time solvable for a graph class whose mim-width is bounded and quickly computable.

For an integer $k \geq 1$ , a $k$ -list assignment of a graph $G = (V, E)$ is a function $L$ that assigns each vertex $u \in V$ a list $L (u) \subseteq {1, 2, \dots, k}$ of admissible colours for $u$ . A colouring $c$ of $G$ respects $L$ if $c (u) \in L (u)$ for every $u \in V$ . For a fixed integer $k \geq 1$ , the List $k$ -Colouring problem is to decide whether a given graph $G$ with a $k$ -list assignment $L$ admits a colouring that respects $L$ . Note that for $k_{1} \leq k_{2}$ , List $k_{1}$ -Colouring is a special case of List $k_{2}$ -Colouring.

[[28]] For every $k \geq 1$ , List $k$ -Colouring is polynomial-time solvable for a graph class whose mim-width is bounded and quickly computable.

In this note, we continue previous work [6] and show that known polynomial-time results for List $k$ -Colouring on special graph classes can be obtained, and strengthened, by applying Theorem 1.

Related Work

A graph is $H$ -free, for some graph $H$ , if it contains no induced subgraph isomorphic to $H$ . For a set of graphs ${H_{1}, \dots, H_{p}}$ , a graph is $(H_{1}, \dots, H_{p})$ -free if it is $H_{i}$ -free for every $i \in {1, \dots, p}$ . We denote the disjoint union of two graphs $G_{1}$ and $G_{2}$ by $G_{1} + G_{2} = (V (G_{1}) \cup V (G_{2}), E (G_{1}) \cup E (G_{2}))$ . We let $P_{r}$ and $K_{r}$ denote the path and complete graph on $r$ vertices, respectively.

The complexity of Colouring for $H$ -free graphs has been settled for every graph $H$ [27], but there are still infinitely many open cases for $k$ -Colouring restricted to $H$ -free graphs when $H$ is a linear forest, that is, a disjoint union of paths. We refer to [15] for a survey and to [8, 10, 25] for updated summaries. In particular, Hoàng et al. [20] proved that for every integer $k \geq 1$ , $k$ -Colouring is polynomial-time solvable for $P_{5}$ -free graphs. This result was generalized by Couturier et al. [11] as follows:

[[11]] For every $k \geq 1$ and $s \geq 0$ , List $k$ -Colouring is polynomial-time solvable for $(s P_{1} + P_{5})$ -free graphs.

We may assume without loss of generality that an instance of List $k$ -Colouring is $K_{k + 1}$ -free, for otherwise it is a no-instance. Hence, Theorem 1 also immediately follows from combining Theorem 1 with the following recent result.

[[6]] For every $r \geq 1$ and $s \geq 0$ , the mim-width of the class of $(K_{r}, s P_{1} + P_{5})$ -free graphs is bounded and quickly computable.

Theorem 1 is part of a recent study [6, 5] on the boundedness of mim-width for hereditary graph classes $G$ characterized by a small set $F_{G}$ of forbidden induced subgraphs. If $F_{G} = {H}$ , then $G$ has bounded mim-width if and only if $G$ has bounded clique-width [5]. On the other hand, this equivalence does not always hold when $F_{G} = {H_{1}, H_{2}}$ [5].

For $r \geq 1$ and $s \geq 1$ , we let $K_{r, s}$ denote the complete bipartite graph with partition classes of size $r$ and $s$ . The graph $K_{1, s}$ is also known as the $(s + 1)$ -vertex star. The $1$ -subdivision of a graph $G$ is the graph obtained from $G$ by subdividing each edge of $G$ exactly once. We denote the $1$ -subdivision of a star $K_{1, s}$ by $K_{1, s}^{1}$ ; in particular $K_{1, 2}^{1} = P_{5}$ .

Even more recently than [6], Chudnovsky, Spirkl and Zhong proved the following result.

[[10]] For every $k \geq 1$ , $s \geq 1$ and $t \geq 1$ , List $3$ -Colouring is polynomial-time solvable for $(K_{1, s}^{1}, P_{t})$ -free graphs.

For every $s \geq 1$ and $t \geq 2 s + 5$ , the class of $(K_{1, s + 2}^{1}, P_{t})$ -free graphs contains the class of $(s P_{1} + P_{5})$ -free graphs. Hence, Theorem 1 generalizes Theorem 1 for the case where $k = 3$ . As $K_{1, s}$ is an induced subgraph of $K_{1, s}^{1}$ , Theorem 1 also generalizes the following result for the case where $r = 1$ .

[[17]] For every $k \geq 1$ , $r \geq 1$ , $s \geq 1$ and $t \geq 1$ , List $k$ -Colouring is polynomial-time solvable for $(K_{r, s}, P_{t})$ -free graphs.

Our Result

In this note we generalize Theorem 1.

For every $r \geq 1$ , $s \geq 1$ and $t \geq 1$ , the mim-width of the class of $(K_{r}, K_{1, s}^{1}, P_{t})$ -free graphs is bounded and quickly computable.

Recall that an instance of List $k$ -Colouring may be assumed to be $K_{k + 1}$ -free. Combining Theorem 1 with Theorem 1 enables us to generalize both Theorems 1 and 1.

For every $k \geq 1$ , $s \geq 1$ and $t \geq 1$ , List $k$ -Colouring is polynomial-time solvable for $(K_{1, s}^{1}, P_{t})$ -free graphs.

Corollary 1 is tight in the following sense. Let $L_{1, s}$ denote the subgraph obtained from $K_{1, s}^{1}$ by subdividing one edge exactly once; in particular $L_{1, 2} = P_{6}$ . Then, as List $4$ -Colouring is NP-complete for $P_{6}$ -free graphs [16], we cannot generalize Corollary 1 to $(L_{1, s}, P_{t})$ -free graphs for $k \geq 4$ , $s \geq 2$ and $t \geq 6$ . Moreover, the mim-width of $(K_{4}, P_{6})$ -free graphs is unbounded [5] and so we cannot extend Theorem 1 to $(K_{r}, L_{1, s}, P_{t})$ -free graphs, for $r \geq 4$ , $s \geq 2$ and $t \geq 6$ , either.

The List $3$ -Colouring problem is polynomial-time solvable for $P_{7}$ -free graphs [4], but the computational complexities of $3$ -Colouring and List $3$ -Colouring is open for $P_{t}$ -free graphs if $t \geq 8$ . In particular, we do not know any integer $t$ such that $3$ -Colouring or List $3$ -Colouring are NP-complete for $P_{t}$ -free graphs. Hence, an extension of Corollary 1 might still be possible for $k = 3$ , and we leave this for future work. This requires more research into the structure of $P_{t}$ -free graphs.

Theorem 1 has other applications as well. For a graph $G$ , let $ω (G)$ denote the size of a maximum clique in $G$ . Chudnovsky et al. [9] gave for the class of $(K_{1, 3}^{1}, P_{6})$ -free graphs, an $n^{O (ω (G)^{3})}$ -time algorithm for Max Partial $H$ -Colouring, a problem equivalent to Independent Set if $H = P_{1}$ and to Odd Cycle Transversal if $H = P_{2}$ . In other words, Max Partial $H$ -Colouring is polynomial-time solvable for $(K_{1, 3}^{1}, P_{6})$ -free graphs with bounded clique number. Chudnovsky et al. [9] noted that Max Partial $H$ -Colouring is polynomial-time solvable for graph classes whose mim-width is bounded and quickly computable. Hence, Theorem 1 generalizes their result for Max Partial $H$ -Colouring to $(K_{1, s}^{1}, P_{t})$ -free graphs with bounded clique number, for any $s \geq 1$ and $t \geq 1$ . However, the running time of the corresponding algorithm is worse than $n^{O (ω (G)^{3})}$ (see [9] for details).

It remains to prove Theorem 1, which we do in the next section.

2 The Proof of Theorem 1

We first state two lemmas. The first lemma shows that given a partition of the vertex set of a graph $G$ , we can bound the mim-width of $G$ in terms of the mim-width of the graphs induced by each part and the mim-width between any two of the parts.

[[6]] Let $G$ be a graph, and let $(X_{1}, \dots, X_{p})$ be a partition of $V (G)$ such that ${c u t m i m}_{G} (X_{i}, X_{j}) \leq c$ for all distinct $i, j \in {1, \dots, p}$ , and $p \geq 2$ . Then

Moreover, if $(T_{i}, δ_{i})$ is a branch decomposition of $G [X_{i}]$ for each $i$ , then we can construct, in $O (1)$ time, a branch decomposition $(T, δ)$ of $G$ with $m i m w (T, δ) \leq max {c ⌊ (\frac{p}{2})^{2} ⌋, {max}_{i \in {1, \dots, p}} {m i m w (T_{i}, δ_{i})} + c (p - 1)}$ .

A clique in a graph is a set of pairwise adjacent vertices. An independent set is a set of pairwise non-adjacent vertices. A dominating set is a set $D$ of vertices such that every vertex not in $D$ is adjacent to at least one vertex in $D$ . For the second lemma we need Ramsey’s Theorem. This theorem states that for every two positive integers $k$ and $ℓ$ , there exists an integer $R (k, ℓ)$ such that every graph on at least $R (k, ℓ)$ vertices contains a clique of size $k$ or an independent set of size $ℓ$ .

For $r \geq 3$ and $s, t \geq 1$ , let $M (r, s, t) = (1 + R (r + 1, R (r + 1, s)))^{t}$ . The next lemma generalizes a result of Chudnovsky, Spirkl and Zhong [10] for the case where $r = 3$ to $r \geq 1$ . The proof of this generalization is analogous to the proof in [10] for the case where $r = 3$ (replace each occurrence of “ $4$ ” in the proofs of Lemma 11 and 13 in [10] by “ $r + 1$ ”).

For every $r \geq 1$ , $s \geq 1$ and $t \geq 1$ , a connected $(K_{r + 1}, K_{1, s}^{1}, P_{t})$ -free graph contains a dominating set of size at most $M (r, s, t)$ .

We are now ready to prove Theorem 1, which we restate below. Its proof closely resembles the proof of Theorem 1. Note that the constant bound depends on $r$ , $s$ and $t$ .

Theorem 1 (restated). For every $r \geq 1$ , $s \geq 1$ and $t \geq 1$ , the mim-width of the class of $(K_{r}, K_{1, s}^{1}, P_{t})$ -free graphs is bounded and quickly computable.

Proof.

Let $G = (V, E)$ be a $(K_{r}, K_{1, s}^{1}, P_{t})$ -free graph for some $r \geq 1$ , $s \geq 1$ and $t \geq 1$ . We may assume without loss of generality that $G$ is connected. We use induction on $r$ . If $r \leq 2$ , then the statement of the theorem holds trivially.

Suppose that $r \geq 3$ . By Lemma 2, we find that $G$ has a dominating set $D$ of size at most $M (r - 1, s, t)$ , which is a constant, as $s$ and $t$ are fixed. Moreover, we can find $D$ in polynomial time by brute force (or we can apply the $O (t n^{2})$ -time algorithm of [10]). We let $p = | D |$ .

We will partition the vertex set of $G$ with respect to $D$ as follows. We first fix an arbitrary ordering $d_{1}, \dots, d_{p}$ on the vertices of $D$ . Let $X_{1}$ be the set of vertices in $V ∖ D$ adjacent to $d_{1}$ . For $i \in {2, \dots, p}$ , let $X_{i}$ be the set of vertices in $V ∖ D$ adjacent to $d_{i}$ , but non-adjacent to any $d_{h}$ with $h \leq i - 1$ . Then $D$ and the sets $X_{1}, \dots, X_{p}$ partition $V$ (some of the sets $X_{i}$ might be empty).

By construction, $d_{i}$ is adjacent to every vertex of $X_{i}$ for each $i \in {1, \dots, p}$ . As $G$ is $K_{r}$ -free, this implies that each $X_{i}$ induces a $(K_{r - 1}, K_{1, s}^{1}, P_{t})$ -free subgraph of $G$ . We denote this subgraph by $G_{i}$ . By the induction hypothesis, the mim-width of $G_{i}$ is bounded and quickly computable for every $i \in {1, \dots, p}$ .

Consider two sets $X_{i}$ and $X_{j}$ with $i < j$ . We claim that ${c u t m i m}_{G} (X_{i}, X_{j}) < c = R (r - 1, R (r - 1, s))$ . Towards a contradiction, suppose that ${c u t m i m}_{G} (X_{i}, X_{j}) \geq c$ . Then, by definition, there exist two sets $A = {a_{1}, a_{2}, \dots, a_{c}} \subseteq X_{i}$ and $B = {b_{1}, b_{2}, \dots, b_{c}} \subseteq X_{j}$ , each of size $c$ , such that ${a_{1} b_{1}, \dots, a_{c} b_{c}}$ is a set of $c$ edges with the property that $G$ does not contain any edges $a_{i} b_{j}$ for $i \neq j$ (note that edges $a_{i} a_{j}$ and $b_{i} b_{j}$ may exist in $G$ ).

As $G [X_{i}]$ is $K_{r - 1}$ -free, Ramsey’s Theorem tells us that the subgraph of $G$ induced by $A$ contains an independent set $A^{'}$ of size $c^{'} = R (r - 1, s)$ . Assume without loss of generality that $A^{'} = {a_{1}, \dots, a_{c^{'}}}$ . Let $B^{'} = {b_{1}, \dots, b_{c^{'}}}$ . As $G [X_{j}]$ is $K_{r - 1}$ -free, the subgraph of $G$ induced by $B^{'}$ contains an independent set $B^{''}$ of size $s$ . Assume without loss of generality that $B^{''} = {b_{1}, \dots, b_{s}}$ . By construction, $d_{i}$ is adjacent to every vertex of ${a_{1}, \dots, a_{s}} \subseteq X_{i}$ and non-adjacent to every vertex of ${b_{1}, \dots, b_{s}} \subseteq X_{j}$ . Hence, ${a_{1}, \dots, a_{s}, b_{1}, \dots, b_{s}, d_{i}}$ induces a $K_{1, s}^{1}$ in $G$ , a contradiction. We conclude that ${c u t m i m}_{G} (X_{i}, X_{j}) < c$ .

We now apply Lemma 2 to find that the mim-width of $G - D$ is bounded and quickly computable. Let $(T, δ)$ be a branch decomposition for $G - D$ with mim-width $k$ . We can readily extend $(T, δ)$ to a branch decomposition $(T^{*}, δ^{*})$ for $G$ with mim-width at most $k + | D | = k + p$ . Namely, we can obtain $T^{*}$ from $T$ by identifying one leaf of $T$ with a leaf of an arbitrary subcubic tree with $p + 2$ leaves. Recall that we found the set $D$ in polynomial time. We conclude that the mim-width of $G$ is bounded and quickly computable. ∎

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List k-Colouring P_t-Free Graphs with No Induced 1-Subdivision of K_1,s: a Mim-width Perspective

Colouring (sP_1+P_5)-Free Graphs: a Mim-Width Perspective

Colouring (P_r+P_s)-Free Graphs

List-three-coloring P_t-free graphs with no induced 1-subdivision of K_1,s

The polynomial method for list-colouring extendability of outerplanar graphs

Colouring Graphs of Bounded Diameter in the Absence of Small Cycles

On algorithmic applications of sim-width and mim-width of (H_1, H_2)-free graphs

Digraphs Homomorphism Problems with Maltsev Condition

1 Introduction

Related Work

Our Result

2 The Proof of Theorem 1

Proof.

References

List k-Colouring P_t-Free Graphs with No Induced 1-Subdivision of K_1,s: a Mim-width Perspective

Related Research

Colouring (sP_1+P_5)-Free Graphs: a Mim-Width Perspective

Colouring (P_r+P_s)-Free Graphs

List-three-coloring P_t-free graphs with no induced 1-subdivision of K_1,s

The polynomial method for list-colouring extendability of outerplanar graphs

Colouring Graphs of Bounded Diameter in the Absence of Small Cycles

On algorithmic applications of sim-width and mim-width of (H_1, H_2)-free graphs

Digraphs Homomorphism Problems with Maltsev Condition

1 Introduction

Related Work

Our Result

2 The Proof of Theorem 1

Proof.

References