$H$-colouring $P_t$-free graphs in subexponential time

TL;DR

This paper demonstrates that the generating function for graph homomorphisms to certain multigraphs and the number of 3-colorings can be computed in subexponential time for $P_t$-free graphs, but 4-colorability remains computationally hard.

Contribution

It introduces a subexponential time algorithm for counting homomorphisms and 3-colorings in $P_t$-free graphs, and establishes a linear bound on pathwidth in terms of maximum degree.

Findings

Subexponential time algorithm for counting (list) homomorphisms to specific multigraphs.

Subexponential time computation of the number of 3-colorings in $P_t$-free graphs.

Hardness result: no subexponential algorithm for 4-colorability assuming ETH.

Abstract

A graph is called $P_t$-free if it does not contain the path on $t$ vertices as an induced subgraph. Let $H$ be a multigraph with the property that any two distinct vertices share at most one common neighbour. We show that the generating function for (list) graph homomorphisms from $G$ to $H$ can be calculated in subexponential time $2^{O\left(\sqrt{tn\log(n)}\right)}$ for $n=|V(G)|$ in the class of $P_t$-free graphs $G$. As a corollary, we show that the number of 3-colourings of a $P_t$-free graph $G$ can be found in subexponential time. On the other hand, no subexponential time algorithm exists for 4-colourability of $P_t$-free graphs assuming the Exponential Time Hypothesis. Along the way, we prove that $P_t$-free graphs have pathwidth that is linear in their maximum degree.

Equations

$$p_{(G,L)\to H}(x):=\sum _{f\in \mathcal{LC}((G,L),H)}\prod _{u\in V(G)}{w}_{u,f(u)}{x}_{f(u)}.$$

$$\prod _{u\in X(i)}{w}_{u,f(u)}{x}_{f(u)}.$$

$$p_{i+1}(g)=\sum _{j=1}^{\ell }{p}_i(g_j)\prod _{v\in X_{i+1}\setminus X_i}{w}_{v,g(v)}{x}_{g(v)}.$$

$$C_h=\{v^{\prime }\in N_G(v):L_{v^{\prime }}\subseteq N_H(h)\}.$$

$$f(n-2)+mf\left(n-y\sqrt{n\log (n)}\right)\le f(n)$$

$$f(n-2)+mf\left(n-y\sqrt{n\log (n)}\right)$$

$$=f(n{)}^{\sqrt{1-2/n}}+mf(n{)}^{{\left(1-y\sqrt{\log (n)/n}\right)}^{\frac{1}{2}}}$$

$$\le f(n{)}^{1-1/n}+mf(n{)}^{1-\frac{1}{2}y\sqrt{\log (n)/n}}$$

$$=f(n)\left[e^{-c\sqrt{\log (n)/n}}+m{e}^{-\frac{1}{2}cy\log (n)}\right]$$

$$e^{-c\sqrt{\log (n)/n}}+m{e}^{-\frac{1}{2}cy\log (n)}\le 1-\frac{1}{2}c\sqrt{\log (n)/n}+m{n}^{-\frac{1}{2}cy}<1.\text{qed}$$

$$f(w-2)+|V(H)|f\left(w-y\sqrt{w\log (w)}\right)$$

$$f(k)+f(\ell )+1$$

$$f(k)+(w-k)+1$$

$$w+1$$

$$\sum _{i=1}^{\ell }1+\sum _{i=\ell +1}^{k}f(w(I_i))+1\le f(w)$$

$$\frac{1}{2}\sqrt{n\log (n)/t}\ge \frac{1}{2\sqrt{t|V(H)|}}\sqrt{w\left(\log (w)-\log (|V(H)|)\right)}\ge y\sqrt{w\log (w)}$$

Full text

††journal: Discrete Applied Mathematics

1 Introduction

Throughout this paper, graphs do not have multiple edges or loops. When we need general graphs (with multiple edges and loops), we call them multigraphs. We use the notation $vv^{\prime}$ for the edge $\{v,v^{\prime}\}$. For a multigraph $G$, the set $N_{G}(v)=\{v^{\prime}:vv^{\prime}\in E(G)\}$ contains $v$ if and only if $G$ has a loop at vertex $v$.

A $k$-colouring of a graph $G$ is a function $c:V(G)\to\{1,\dots,k\}$ such that $c(v)\neq c(v^{\prime})$ for all $vv^{\prime}\in E(G)$. The decision problem $k$-colourability asks whether a given graph $G$ is $k$-colourable. This problem is NP-complete for $k\geq 3$ in general. In order to investigate what graph structure makes the decision problem hard, a natural question is whether the problem becomes easy if it is restricted to instances that do not contain a particular structure. Thus we restrict to the class of $F$-free graphs, that is, those graphs which do not contain $F$ as an induced subgraph, for some fixed graph $F$. If $k$ is part of the input, a full classification is given by Král et al. [18]. For fixed $k\geq 3$, $k$-colourability is shown to be NP-complete for $F$-free graphs if $F$ is either a cycle $C_{\ell}$ for $\ell\geq 3$ [17] or the claw $K_{1,3}$ [14, 19]. Any graph $F$ which does not contain a cycle nor the claw is a disjoint union of paths.

Let $P_{t}$ denote the path on $t$ vertices. Polynomial-time algorithms for deciding $k$-colourability for $P_{t}$-free graphs exist for $t\leq 5$ [13], $(k,t)=(4,6)$ [6, 7] and $(k,t)=(3,7)$ [4]. On the other hand, Huang [15] showed $4$-colourability is NP-complete for $P_{7}$-free graphs and $5$-colourability is NP-complete for $P_{6}$-free graphs. It is an open problem to determine the complexity of 3-colourability of $P_{t}$-free graphs for $t\geq 8$.

It is also open whether maximum independent set is decidable in polynomial time for $P_{t}$-free graphs for $t\geq 7$. Brause [5] and Bascó, Marx and Tuza [2] independently showed a greedy exhaustive approach yields a subexponential-time algorithm for maximum independent set on $P_{t}$-free graphs. In this paper we show that there are subexponential-time algorithms for a larger class of problems, including $3$-colourability and maximum independent set, and also give counting results. Our algorithm builds on the following property of $P_{t}$-free graphs.

Lemma 1**.**

A $P_{t}$-free graph of maximum degree $\Delta$ has pathwidth at most $(\Delta-1)(t-2)+1$. Moreover, a path-decomposition of this width can be found in polynomial time.

This lemma is an improvement on the treewidth bound of Bascó et al. [1] for $P_{t}$-free graphs (which they used to improve the algorithm of Bascó et al. [2]). The result is tight up to a constant factor, which can be seen by replacing the vertices of a binary tree by cliques.

We now introduce our framework. Let $H$ be a multigraph and $G$ a (simple) graph. A graph homomorphism from $G$ to $H$ is a map $f:V(G)\to V(H)$ such that $vv^{\prime}\in E(G)$ implies $f(v)f(v^{\prime})\in E(H)$. (Thus a 3-colouring is a graph homomorphism to $K_{3}$.) A list $H$-colouring instance $I=(G,L)$ consists of a graph $G$ together with a function $L:V(G)\to\mathcal{P}(V(H))$ that assigns a subset $L_{v}\subseteq V(H)$ to every $v\in V(G)$. A list $H$-colouring of such an instance is a graph homomorphism $f$ from $G$ to $H$ such that $f(v)\in L_{v}$ for all $v\in V(G)$. We denote the set of list $H$-colourings of $(G,L)$ by $\mathcal{LC}((G,L),H)$.

A useful way to summarise information about $H$-colourings of a graph $G$ is to use a multivariate generating function (see for example [21]). Given a multigraph $H$, we define the partition function $p_{(G,L)\to H}(x)$ by

[TABLE]

We omit the lists $L$ where clear from context. The weights $w_{v,h}$ (for $v\in V(G)$ and $h\in V(H)$) are included to allow more general application and can be ignored by choosing them identically one. For $H=(\{h,h^{\prime}\},\{hh^{\prime},h^{\prime}h^{\prime}\})$, $p_{G\to H}(x)$ gives the independent set polynomial when $x_{h^{\prime}}$ is set to one.

Summing appropriate coefficients of the polynomial, the partition function can be used to for example count the number of list $H$-colourings, or to count the number of “restrictive $H$-colourings” [8] in which a restriction is placed on the size of the preimages of the vertices of $H$.

The following theorem is our main result and will be proved in Section 3.

Theorem 2**.**

Let $H$ be a multigraph so that $|N_{H}(h)\cap N_{H}(h^{\prime})|\leq 1$ for all distinct vertices $h,h^{\prime}$ of $H$. For $t\geq 4$, the polynomial $p_{G\to H}(x)$ can be calculated for every $P_{t}$-free graph $G$ in time $2^{O\left(\sqrt{tn\log(n)}\right)}$ where $n=|V(G)|$.

For simple graphs $H$, the condition $|N_{H}(h)\cap N_{H}(h^{\prime})|\leq 1$ for all distinct $h,h^{\prime}$ is equivalent to $H$ not having $C_{4}$ as (not necessarily induced) subgraph.

Corollary 3**.**

The following problems can be solved for $P_{t}$-free graphs $G$ in time $2^{O\left(\sqrt{tn\log(n)}\right)}$ where $n=|V(G)|$:

• •

Counting the number of $H$-colourings for any fixed simple graph $H$ with no $C_{4}$ subgraph.

• •

Computing the independent set polynomial.

In particular, it can be decided in subexponential time whether a $P_{t}$-free graph is 3-colourable (and the number of 3-colourings can be counted).

For example, if $G$ is $P_{t}$-free and $H$ is an odd cycle, then we can count the number of $H$-colourings of $G$ in subexponential time. This problem is $\#P$-complete if all graphs $G$ are allowed [9] and the corresponding decision problem is NP-complete [12].

The Exponential Time Hypothesis (ETH) [16] states that there is an $\epsilon>0$ such that there is no $O(2^{\epsilon n})$-time algorithm for 3-sat. In Section 4 we prove the following result that shows that it is in some sense unlikely that Corollary 3 will extend to $k$-colouring $P_{t}$-free graphs for $k>3$.

Proposition 4**.**

If the Exponential Time Hypothesis is true, then there is no algorithm running in time subexponential in the number of vertices of the graph for

• •

$4$-colourability* on $P_{7}$-free graphs;*

• •

$3$-colourability* for $F$-free graphs for any connected $F$ which is not a path.*

This result might shed some light on why the complexity status of $3$-colourability of $P_{t}$-free graphs for $t$ large has remained open whereas the complexity of $k$-colourability of $F$-free graphs has been settled for other values of $k$ or $F$.

2 Pathwidth of $P_{t}$-free graphs and dynamic programming

A path-decomposition of a graph $G$ is a sequence of subsets $X_{i}$ of vertices of $G$ with three properties:

• •

The vertex set of $G$ equals $\bigcup_{i}X_{i}$,

• •

For each edge of $G$, there exists an $i$ such that both endpoints of the edge belong to subset $X_{i}$, and

• •

$X_{\ell}\cap X_{j}\subseteq X_{i}$ for every three indices $1\leq\ell\leq i\leq j$.

The pathwidth of $G$ is defined as the minimum of $\max_{i}|X_{i}|-1$ over the path-decompositions of $G$.

Let $T$ be a rooted tree with root $r$. For a vertex $v$, let $T_{v}$ denote the path of $T$ between $r$ and $v$. For each vertex $w$, fix a linear order of the set of children of $w$. We call this a plane tree. If $u,v$ are both children of $w$ and $u$ is earlier than $v$ in the corresponding ordering, we say $u$ is an elder sibling of $v$.

Let $T$ be a spanning tree of a graph $G$, equipped with orders to make it a plane tree. We call it an uncle tree of $G$ if for every edge $uv$ of $G$ that is not an edge of $T$, one of $u,v$ has an elder sibling that is an ancestor of the other. We use the following result of Seymour [22].

Theorem 5**.**

For every connected graph we can compute an uncle tree with any specified vertex as root in polynomial time.

(The proof is easy; grow a depth-first tree, subject to the condition that the path of the tree between each vertex and the root is induced.)

Proof of Lemma 1.

We may assume that $G$ is connected. Take an uncle tree $T$ of $G$, and order its leaves as $p_{1},\ldots,p_{k}$ say, in the natural order of the leaves of a plane tree. For $1\leq i\leq k$, let $X_{i}$ be the set of vertices of $T$ that either belong to $T_{p_{i}}$, or have an elder sibling (and hence also a parent) in this set. We claim that the sequence $(X_{1},\ldots,X_{k})$ is a path-decomposition of $G$. We check that:

• •

Every vertex belongs to some $X_{i}$ (this is clear).

• •

For all $u,v$ adjacent in $G$, there exists $i$ with $u,v\in X_{i}$. To see this, since $T$ is an uncle tree we may assume that $u$ has an elder sibling $u^{\prime}$ that is an ancestor of $v$. Choose a leaf $p_{i}$ of $T$ such that $T_{p_{i}}$ contains $v$; then the common parent of $u,u^{\prime}$ belongs to $T_{p_{i}}$, and hence $u,v\in X_{i}$.

• •

$X_{\ell}\cap X_{j}\subseteq X_{i}$ for $1\leq\ell\leq i\leq j\leq k$. To see this, let $v\in X_{\ell}\cap X_{j}$; consequently either $v$ or an elder sibling of $v$ belongs to $T_{p_{\ell}}$, and either $v$ or an elder sibling belongs to $T_{p_{j}}$. It follows that either $v$ or an elder sibling of $v$ belongs to $T_{p_{i}}$, from the ordering of the leaves of $T$.

This proves the claim.

If $T$ is an uncle tree of $G$, then each path of $T$ starting from $r$ is induced in $G$; and so if $G$ does not contain $P_{t}$, then $|T_{p_{i}}|\leq t-1$ for each $i$, and so $|X_{i}|\leq\Delta-1+(t-3)(\Delta-2)+1$, where $\Delta$ denotes the maximum degree of $G$. ∎

We give a short outline of how the standard dynamic programming approach can be applied to compute $p_{G\to H}(x)$ in time $2^{O(p)}n$ given a path-decomposition of width $p$ of a graph $G$ on $n$ vertices.

Let $(X_{1},\dots,X_{k})$ be the given path-decomposition of $G$. We define $X(i)=\bigcup_{j\leq i}X_{j}$. For each $i\in[k]$ and list $H$-colouring $g:X_{i}\to V(H)$, we compute the polynomial $p_{G[X(i)]\to H}$ with the vertices in $X_{i}$ precoloured: we define $p_{i}(g)$ as the sum, over the list $H$-colourings $f:X(i)\to V(H)$ such that $f|_{X_{i}}=g$, of

[TABLE]

For each list $H$-colouring $g:X_{1}\to V(H)$, we set $p_{1}(g)=\prod_{v\in X_{1}}w_{v,g(v)}x_{g(v)}$. Having computed all $p_{i}(g)$ for some $i$, for each list $H$-colouring $g:X_{i+1}\to V(H)$ we select the list $H$-colourings $g_{1},\dots,g_{\ell}$ of $X_{i}$ that are compatible with $g$, that is, $g_{i}(v)=g(v)$ for $v\in X_{i}\cap X_{i+1}$ and $g(u)g(v)\in E(H)$ if $uv\in E(G)$ for all $u\in X_{i}$ and $v\in X_{i+1}$. Since $N_{G}[v]\cap X(i)\subseteq X_{i}$ for all $v\in X_{i+1}\setminus X_{i}$, we can then compute

[TABLE]

Finally, we calculate the desired $p_{G\to H}$, which is the sum, over all list $H$-colourings $g$ of $X_{k}$, of $p_{k}(g)$.

3 Algorithm and time analysis

Throughout this section, $H$ is a fixed multigraph such that $|N_{H}(h)\cap N_{H}(h^{\prime})|\leq 1$ for all distinct $h,h^{\prime}$ in $H$. We allow loops in $H$ but no multiple edges.333It is possible to extend the algorithm to compute a version of $p_{G\to H}(x)$ with edge weights $A_{h,h^{\prime}}$ for $h,h^{\prime}\in V(H)$, but in this case the weights $w_{v^{\prime},h^{\prime}}$ have to be updated in Line 3 and 6 of Algorithm HCol to $w_{v^{\prime},h^{\prime}}A_{h,h^{\prime}}$ for all $v^{\prime}\in N_{v}$. The $P_{t}$-free graphs $G$ are assumed to be simple.

We shall say a list colouring instance $I=(G,L)$ has weight $w(I)=\sum_{v\in V(G)}|L_{v}|$ and is reduced if $|L_{v}|\geq 2$ for all $v\in V(G)$. The key observation we need is the following.

Lemma 6**.**

Let $I=(G,L)$ be a reduced list $H$-colouring instance and let $v\in V(G)$ with degree $d(v)$. For $h\in V(H)$, let

[TABLE]

Then there is at most one $h\in L_{v}$ for which $|C_{h}|>\frac{1}{2}d(v)$.

Proof.

Suppose $h\neq h^{\prime}$ in $L_{v}$ both satisfy $|C_{h}|,|C_{h^{\prime}}|>\frac{1}{2}d(v)$. Then there exists $v^{\prime}\in N_{G}(v)$ such that $v^{\prime}\in C_{h}\cap C_{h^{\prime}}$. Hence $L_{v^{\prime}}\subseteq N_{H}(h)\cap N_{H}(h^{\prime})$, so that by our assumption on $H$ we find $|L_{v^{\prime}}|\leq 1$, contradicting the assumption that $I$ is reduced. ∎

This lemma tells us that “colouring” a vertex $v$ of degree $d(v)=\Delta$ decreases the weight of a reduced instance by at least $\frac{1}{2}\Delta$ for all but one “colour” in $L_{v}$. Either there is a vertex of high degree and we can reduce the weight significantly by colouring this vertex, or $\Delta$ is “small” and we can apply the results from the previous section to compute $p_{G\to H}(x)$ in time $2^{O(t\Delta)}$.

Our algorithm “HCol” for computing the list $H$-colouring function of a graph $G$ is given below. This algorithm either terminates or recurses on instances of strictly smaller weight. Therefore, it always terminates in finite time. We can represent the recursions by a tree: the root is the first call of the algorithm and each recursive call creates a child. For $P_{t}$-free graphs, we can bound the number of nodes in this recursion tree.

Proposition 7**.**

Let $t\geq 4$, $c>4\sqrt{t|V(H)|}/\log(2)$ and $f(w)=2^{c\sqrt{w\log(w)}}$. Then there exists an $n_{0}$ for Algorithm HCol such that if it is applied to an instance $I=(G,L)$ of weight $w(I)$ with $G$ a $P_{t}$-free graph, then the number of nodes in the corresponding recursion tree is bounded by $f(w(I))$.

Algorithm HCol gives the correct answer for any graph $G$: in line 2 we note that if some vertex has an empty list, then $p_{G\to H}=0$; in line 3 we note that if the list of a vertex $v\in V(G)$ has a single element $h\in V(H)$, then $v$ has to be mapped to $h$, i.e. $p_{G\to H}(x)=w_{v,h}x_{h}p_{G-v\to H}(x)$; in line 4 we use the algebraic identity $p_{G_{1}\sqcup G_{2}\to H}(x)=p_{G_{1}\to H}(x)p_{G_{2}\to H}(x)$; in line 6, we use $p_{G\to H}(x)=\sum_{h\in L_{v}}w_{v,h}x_{h}p_{G-{v}\to H}(x)$.444We left the lists of the vertices implicit in the notation; the precise way in which the lists need to be updated is given in the algorithm.

Inspecting and updating the lists of vertices, finding a vertex of maximal degree and finding the connected components of a graph on $n$ vertices can all be done in time $Cn^{2}$, where the constant $C$ may depend on $H$ and $n_{0}$. Line 5 is applied at most once per node and takes $2^{O\left(\sqrt{tn\log(n)}\right)}$. Since $|L_{v}|\leq|V(H)|$ for all $v\in V(G)$, it follows that $w(I)\leq|V(H)|n=O(n)$. Theorem 2 follows from Proposition 7 by observing that $Cn^{2}2^{O\left(\sqrt{tn\log(n)}\right)}2^{O\left(\sqrt{tn\log(n)}\right)}=2^{O\left(\sqrt{tn\log(n)}\right)}$.

We require the following simple estimate.

Lemma 8**.**

Let $m,y>0$ and $c>y^{-1}$. There exists an $n_{0}\in\mathbb{N}$ such that $f(w)=e^{c\sqrt{w\log(w)}}$ satisfies

[TABLE]

for all $n\geq n_{0}$.

Proof.

Let $\epsilon>0$ be given such that $\sqrt{1-x}\leq 1-\frac{1}{2}x$ and $e^{-x}\leq 1-x/2$ for all $0\leq x\leq\epsilon$. Choose $n_{0}$ sufficiently large such that $2/n,y\sqrt{\log(n)/n},c\log(n)/\sqrt{n}\leq\epsilon$ and $mn^{-cy/2}<\frac{c}{2}\sqrt{\log(n)/n}$ for all $n\geq n_{0}$ (by assumption, $cy>1$). We calculate

[TABLE]

But

[TABLE]

Proof of Proposition 7.

Let $n_{0}$ be given from Lemma 8 applied with $m=|V(H)|$, $y=\frac{1}{4\sqrt{t|V(H)|}}$ and using $c\log(2)$ instead of $c$. Enlarging $n_{0}$ if necessary for the last three properties, we may now assume that

[TABLE]

The proposition is proved by induction on $w=w(I)$. If the algorithm terminates in line 1, 2 or 5, then there is only one iteration and $f(n)\geq 1$ for all $n\in\mathbb{Z}_{\geq 0}$. If the algorithm reaches line 3, then $G$ has at least $n_{0}$ vertices and $|L_{v}|\geq 1$ for all $v\in V$, so that $w(I)\geq n_{0}$. Therefore, the statement holds for all $w<n_{0}$.

Suppose the statement has been shown for instances with $w(I)<w$ for some $w\geq n_{0}$. If the algorithm recurses on line 3, then the removed vertex contributed at least 1 to the weight, and so by induction at most $f(w-1)+1\leq f(w)$ iterations are taken.

If the algorithm reaches line 4, we may assume the instance is reduced, and so $w(I)\leq 2|V(G)|$. Suppose the graph $G$ is disconnected with connected components $G_{1},\dots,G_{k}$. Let $I_{i}=(G_{i},L|_{V(G_{i})})$ and note that $I_{i}$ is also reduced. Hence if $|V(G_{i})|\leq\frac{1}{2}w(I_{i})\leq n_{0}$, then the recursive call on $G_{i}$ will take a single iteration. Renumber so that $I_{1},\dots,I_{\ell}$ have weight at most $2n_{0}$ and $I_{\ell+1},\dots,I_{k}$ have weight at least $2n_{0}$. By induction the algorithm takes at most

[TABLE]

iterations, where the inequality follows from the assumptions we placed on $n_{0}$, considering $k-\ell=0$, $k-\ell=1$ and $k-\ell>1$ separately.

At line 6, the hypothesis of Lemma 6 is satisfied. All but one of the instances have their weight reduced by at least

[TABLE]

(using that $w=\sum_{v\in V(G)}|L_{v}|\leq|V(H)|n$ and assuming $\log(w)-\log(|V(H)|)\geq\frac{1}{4}\log(w)$ by enlarging $n_{0}$ if necessary). The other instance has its weight reduced by at least 2, since the vertex $v$ with $|L_{v}|\geq 2$ is removed. By induction, the number of nodes in the recursion tree is at most $f(w-2)+(|V(H)|-1)f\left(w-y\sqrt{w\log(w)}\right)+1\leq f(w)$. ∎

4 Extensions

Our algorithm can easily be adapted to find the $H$-colouring $f$ of $G$ which minimises the cost $\sum_{v\in V(G)}w_{v,f(v)}$; in particular, Minimum Cost Homomorphism [11] can be solved in subexponential time for $G$ and $H$ as above.

Note that Lemma 6 is the bottleneck for extending the time complexity to, for example, $4$-colouring ($H=K_{4}$): it is possible that the neighbourhood of a vertex $v$ with $L_{v}=\{1,2,3,4\}$ has mostly neighbours with list $\{1,2\}$, so that both $C_{3}$ and $C_{4}$ are large. Assuming the Exponential Time Hypothesis (ETH), this is to be expected in view of Proposition 4: under ETH, no such subexponential time algorithm can exist for $4$-colourability on $P_{7}$-free graphs.

Proof of Proposition 4.

Huang [15] gives a reduction of an instance of 3-sat with $m$ formulas and $n$ variables into an instance of 4-colourability for a $P_{7}$-free graph on $O(n+m)$ vertices. Therefore, any algorithm for 4-colourability on $P_{7}$-free graphs yields an algorithm for 3-sat with the same time dependence on the input size.

Let $F$ be a connected graph which is not a path. Then $F$ contains either the claw $K_{1,3}$ or a cycle. We prove ETH implies that there is no subexponential time algorithm for $3$-colourability of $F$-free graphs. The standard reduction (for example [10, Prop. 2.26]) from 3-sat to 3-colourability creates a graph on $O(n+m)$ vertices. Kamiński and Lozin [17] reduce 3-colourability to 3-colourability on graphs of girth at least $g$ (for every $g\geq 3$) by (in the worst case) replacing each vertex by a constant-sized gadget. This handles the case when $F$ contains a cycle.

For $F$ containing the claw $K_{1,3}$, note that claw-free graphs are a superset of line graphs. Holyer [14] reduces 3-sat to 3-edge colourability on $3$-regular graphs. Given $n$ variables and $m$ clauses, constant-sized gadgets are created for each variable and clause; additional components are added to the variable gadgets for each time it occurs in a clause. Since there are at most $3m$ such occurrences, this creates a graph on $O(n+m)$ vertices. Since the graph is 3-regular, the number of edges is $O(n+m)$ as well. Hence the line graph of such a graph will have $O(n+m)$ vertices. Since $3$-colourings of the vertices of the line graph are in one-to-one correspondence with $3$-colourings of the edges of the original graph, we can hence reduce 3-sat to 3-colourability of line graphs on $O(n+m)$ vertices. ∎

If ETH holds, then any polynomial-time reduction from 3-sat to a problem with a subexponential algorithm must “blow up” the instance size. Our result therefore suggests that, if one attempts to prove NP-completeness of 3-colourability for $P_{t}$-free graphs (for $t$ large) by designing gadgets, it may be necessary either to start from a problem whose instance size has already been “blown up” or to use gadgets which are not of bounded size.

Our algorithm only uses the property of $P_{t}$-free graphs that every induced subgraph has pathwidth $O(t\Delta)$. Seymour [22] proves that a tree-decomposition of width $O(\ell\Delta)$ can be computed efficiently for graphs that do not contain cycles of length at least $\ell$ as induced subgraph; therefore, our algorithm extends to this class of graphs (after adjusting the standard dynamic programming approach of for example [3] to our setting in a similar fashion as done in Section 2). This motivates the following question.

Problem**.**

For fixed $t$, are 3-colourability and maximum independent set solvable in polynomial time on graphs that have no induced cycles of length greater than $t$?

Theorem 5 can also be used to bound tree-depth of $P_{t}$-free graphs. The tree-depth of a graph $G$ is the minimum height of a forest $F$ on the same vertex set with the property that for every edge of $G$, the corresponding vertices are in an ancestor-descendant relationship to each other in $F$ [20].

Corollary 9**.**

The tree-depth of a connected $P_{t}$-free graph $G$ of maximum degree $\Delta$ is at most $(t-2)(\Delta-1)+1$.

Such a desired forest $F$ can be computed as follows. First compute an uncle tree $T$ for $G$. For each non-leaf vertex $v\in T$, create a path $P_{v}$ containing all the children of $v$. The forest $F$ is obtained by connecting the “end”point of a path $P_{v}$ to the “start”points of the paths of its children. Now recall that an uncle tree has height at most $t-1$ since each path from the root to a leaf is induced and that a non-root, non-leaf node has at most $\Delta-1$ children.