Fine-grained complexity of the graph homomorphism problem for bounded-treewidth graphs

TL;DR

This paper investigates the computational complexity of the graph homomorphism problem for graphs with bounded treewidth, establishing tight bounds under the Strong Exponential Time Hypothesis for a broad class of graphs.

Contribution

It provides a complexity classification for Hom(H) on bounded-treewidth graphs, especially for projective core graphs, and connects algebraic graph theory conjectures to algorithmic limits.

Findings

No faster algorithms exist for projective core graphs under SETH.

The naive dynamic programming algorithm is optimal for most graphs.

The classification covers all known cases, linking algebraic conjectures to complexity bounds.

Abstract

For graphs $G$ and $H$, a \emph{homomorphism} from $G$ to $H$ is an edge-preserving mapping from the vertex set of $G$ to the vertex set of $H$. For a fixed graph $H$, by \textsc{Hom($H$)} we denote the computational problem which asks whether a given graph $G$ admits a homomorphism to $H$. If $H$ is a complete graph with $k$ vertices, then \textsc{Hom($H$)} is equivalent to the $k$-\textsc{Coloring} problem, so graph homomorphisms can be seen as generalizations of colorings. It is known that \textsc{Hom($H$)} is polynomial-time solvable if $H$ is bipartite or has a vertex with a loop, and NP-complete otherwise [Hell and Ne\v{s}et\v{r}il, JCTB 1990]. In this paper we are interested in the complexity of the problem, parameterized by the treewidth of the input graph $G$. If $G$ has $n$ vertices and is given along with its tree decomposition of width $\mathrm{tw}(G)$, then the problem can be solved in time $|V(H)|^{\mathrm{tw}(G)} \cdot n^{\mathcal{O}(1)}$, using a straightforward dynamic programming. We explore whether this bound can be improved. We show that if $H$ is a \emph{projective core}, then the existence of such a faster algorithm is unlikely: assuming the Strong Exponential Time Hypothesis (SETH), the \textsc{Hom($H$)} problem cannot be solved in time $(|V(H)|-\epsilon)^{\mathrm{tw}(G)} \cdot n^{\mathcal{O}(1)}$, for any $\epsilon > 0$. This result provides a full complexity characterization for a large class of graphs $H$, as almost all graphs are projective cores. We also notice that the naive algorithm can be improved for some graphs $H$, and show a complexity classification for all graphs $H$, assuming two conjectures from algebraic graph theory. In particular, there are no known graphs $H$ which are not covered by our result. In order to prove our results, we bring together some tools and techniques from algebra and from fine-grained complexity.

Equations

$$V(H_1\times H_2)=$$

$$E(H_1\times H_2)=$$

$$H_1\times \cdots \times H_{m-1}\times H_m:=(H_1\times \cdots \times H_{m-1})\times H_m.$$

$$G\to H\text{ if and only if }G\to H_i\text{ for every }i\in [m].$$

$$g(x)=\{\begin{array}{ll}f(x)& \text{if x∈V(G′),}\\ {\pi }_j(x)& \text{otherwise.}\end{array}$$

$$u^{\ast }:=(z_1^{k-1},\dots ,z_k^{k-1})\text{ and }{v}^{\ast }:=(\overline{z_1},\dots ,\overline{z_k})$$

$$\{y\in V(H)\text{leavevmode}|\text{leavevmode}\exists f:K\to H\text{ such that }f(x_i)=y_i\text{ for every }i\in [\ell ]\text{ and }f(x_0)=y\}=C.$$

$$f(x)=(f_1(x),\dots ,f_m(x)),$$

$$u:=$$

$$v:=$$

$$(\mu \circ f)(z,\dots ,z,r)=(\sigma \circ g\circ f)(z,\dots ,z,r)=(\sigma \circ \mathrm{id})(z,\dots ,z,r)=\sigma (z,\dots ,z,r)=(z,r),$$

$$(\mu \circ f{)}_1(u^{\ast })(\mu \circ f{)}_1(v^{\ast })\in E(H_1).$$

$$\displaystyle\linenomath\begin{split}\left(\mu\circ f\right)(\bar{z})=\mu\Big{(}f_{1}(\bar{z}),f_{2}(\bar{z})\Big{)}=\Bigg{(}\mu_{1}\Big{(}f_{1}(\bar{z}),f_{2}(\bar{z})\Big{)},\mu_{2}\Big{(}f_{1}(\bar{z}),f_{2}(\bar{z})\Big{)}\Bigg{)}.\end{split}\endlinenomath$$

$$\displaystyle\linenomath\begin{split}\mu_{1}\big{(}f_{1}(\bar{z}),f_{2}(\bar{z})\big{)}=&\mu^{(1)}(f_{1}(\bar{z}))\\ \mu_{2}\big{(}f_{1}(\bar{z}),f_{2}(\bar{z})\big{)}=&\mu^{(2)}(f_{2}(\bar{z})),\end{split}\endlinenomath$$

$$\left({\mu }^{(1)}\circ f_1\right)(u^{\ast })\left({\mu }^{(1)}\circ f_1\right)(v^{\ast })\in E(H_1).$$

Full text

Fine-grained complexity of the graph homomorphism problem

for bounded-treewidth graphs††thanks: The extended abstract of this work was presented during the conference SODA 2020 [40]

Karolina Okrasa E-mail: k.okrasa@mini.pw.edu.pl. Supported by the ERC grant CUTACOMBS (no. 714704). Faculty of Mathematics and Information Science, Warsaw University of Technology, Poland

Faculty of Mathematics, Informatics and Mechanics, University of Warsaw

Paweł Rzążewski E-mail: p.rzazewski@mini.pw.edu.pl. Supported by Polish National Science Centre grant no. 2018/31/D/ST6/00062. Faculty of Mathematics and Information Science, Warsaw University of Technology, Poland

Faculty of Mathematics, Informatics and Mechanics, University of Warsaw

For graphs $G$ and $H$, a homomorphism from $G$ to $H$ is an edge-preserving mapping from the vertex set of $G$ to the vertex set of $H$. For a fixed graph $H$, by Hom($H$) we denote the computational problem which asks whether a given graph $G$ admits a homomorphism to $H$. If $H$ is a complete graph with $k$ vertices, then Hom($H$) is equivalent to the $k$-Coloring problem, so graph homomorphisms can be seen as generalizations of colorings. It is known that Hom($H$) is polynomial-time solvable if $H$ is bipartite or has a vertex with a loop, and NP-complete otherwise [Hell and Nešetřil, JCTB 1990].

In this paper we are interested in the complexity of the problem, parameterized by the treewidth of the input graph $G$. If $G$ has $n$ vertices and is given along with its tree decomposition of width $\mathrm{tw}(G)$, then the problem can be solved in time $|V(H)|^{\mathrm{tw}(G)}\cdot n^{\mathcal{O}(1)}$, using a straightforward dynamic programming. We explore whether this bound can be improved. We show that if $H$ is a projective core, then the existence of such a faster algorithm is unlikely: assuming the Strong Exponential Time Hypothesis (SETH), the Hom($H$) problem cannot be solved in time $(|V(H)|-\varepsilon)^{\mathrm{tw}(G)}\cdot n^{\mathcal{O}(1)}$, for any $\varepsilon>0$. This result provides a full complexity characterization for a large class of graphs $H$, as almost all graphs are projective cores.

We also notice that the naive algorithm can be improved for some graphs $H$, and show a complexity classification for all graphs $H$, assuming two conjectures from algebraic graph theory. In particular, there are no known graphs $H$ which are not covered by our result.

In order to prove our results, we bring together some tools and techniques from algebra and from fine-grained complexity.

1 Introduction

Many problems that are intractable for general graphs become significantly easier if the structure of the input instance is “simple”. One of the most successful measures of such a structural simplicity is the treewidth of a graph, whose notion was rediscovered by many authors in different contexts [3, 22, 42, 1]. Most classic NP-hard problems, including Independent Set, Dominating Set, Hamiltonian Cycle, or Coloring, can be solved in time $\mathcal{O}^{*}(f({\operatorname{tw}(G)}))$, where ${\operatorname{tw}(G)}$ is the treewidth of the input graph $G$ (in the $\mathcal{O}^{*}(\cdot)$ notation we suppress factors polynomial in the input size) [2, 7, 11, 13]. In other words, many problems become polynomially solvable for graphs with bounded treewidth.

In the past few years the notion of fine-grained complexity gained popularity, and the researchers became interested in understanding what is the optimal dependence on the treewidth, i.e., the function $f$ in the complexity of algorithms solving particular problems. This led to many interesting algorithmic results and lower bounds [49, 6, 31, 35, 41, 13]. Note that the usual assumption that P $\neq$ NP is not strong enough to obtain tight bounds for the running times of algorithms. In the negative results we usually assume the Exponential Time Hypothesis (ETH), or the Strong Exponential Time Hypothesis (SETH) [29, 30]. Informally speaking, the ETH asserts that 3-Sat with $n$ variables and $m$ clauses cannot be solved in time $2^{o(n+m)}$, while the SETH implies that CNF-Sat with $n$ variables and $m$ clauses cannot be solved in time $(2-\varepsilon)^{n}\cdot m^{\mathcal{O}(1)}$, for any $\varepsilon>0$.

For example, it is known that for every fixed $k$, the $k$-Coloring problem can be solved in time $\mathcal{O}^{*}(k^{{\operatorname{tw}(G)}})$, if a tree decomposition of $G$ of width ${\operatorname{tw}(G)}$ is given [5, 13]. On the other hand, Lokshtanov, Marx, and Saurabh showed that this result is essentially optimal, assuming the SETH.

Theorem 1** (Lokshtanov, Marx, Saurabh [36]).**

*Let $k\geqslant 3$ be a fixed integer. Assuming the SETH, the $k$-Coloring problem on a graph $G$ cannot be solved in time $\mathcal{O}^{*}\left((k-\varepsilon)^{{\operatorname{tw}(G)}}\right)$ for any $\varepsilon>0$. *

Homomorphisms

For two graphs $G$ and $H$, a homomorphism is an edge-preserving mapping from $V(G)$ to $V(H)$. The graph $H$ is called the target of the homomorphism. The existence of a homomorphism from any graph $G$ to the complete graph $K_{k}$ is equivalent to the existence of a $k$-coloring of $G$. Because of that we often refer to a homomorphism to $H$ as an $H$-coloring and think of vertices of $H$ as colors. We also say that a graph $G$ is $H$-colorable if it admits a homomorphism to $H$. For a fixed graph $H$, by Hom($H$) we denote the computational problem which asks whether a given instance graph $G$ admits a homomorphism to $H$. Clearly Hom($K_{k}$) is equivalent to $k$-Coloring.

Since $k$-Coloring is arguably one of the best studied computational problems, it is interesting to investigate how these results generalize to Hom($H$) for non-complete targets $H$. For example, it is known that $k$-Coloring is polynomial-time solvable for $k\leqslant 2$, and NP-complete otherwise. A celebrated result by Hell and Nešetřil [26] states that Hom($H$) is polynomially solvable if $H$ is bipartite or has a vertex with a loop, and otherwise is NP-complete. The polynomial part of the theorem is straightforward and the main contribution was to prove hardness for all non-bipartite graphs $H$. The difficulty comes from the fact that the local structure of the graph $H$ is not very helpful, but we need to consider $H$ as a whole. This is the reason why the proof of Hell and Nešetřil uses a combination of combinatorial and algebraic arguments. Several alternative proofs of the result have appeared [10, 46], but none of them is purely combinatorial.

When it comes to the running times of algorithms for $k$-Coloring, it is well-known that the trivial $\mathcal{O}^{*}(k^{n})$ algorithm for $k$-Coloring, where $n$ is the number of vertices of the input graph, can be improved to $\mathcal{O}^{*}(c^{n})$ for a constant $c$ which does not depend on $k$ (currently the best algorithm of this type has running time $\mathcal{O}^{*}(2^{n})$ [4]). Analogously, we can ask whether the trivial $\mathcal{O}^{*}(|H|^{n})$ algorithm for Hom($H$) can be improved, where by $|H|$ we mean the number of vertices of $H$. There are several algorithms with running times $\mathcal{O}^{*}({c(H)}^{n})$, where $c(H)$ is some structural parameter of $H$, which could be much smaller than $|H|$ [20, 50, 43]. However, the question whether there exists an absolute constant $c$, such that for every $H$ the Hom($H$) problem can be solved in time $\mathcal{O}^{*}(c^{n})$, remained open. Finally, it was answered in the negative by Cygan et al. [12], who proved that the $\mathcal{O}^{*}(|H|^{n})$ algorithm is essentially optimal, assuming the ETH.

Using a standard dynamic programming approach, Hom($H$) can be solved in time $\mathcal{O}^{*}(|H|^{t})$, if an input graph is given along with its tree decomposition of width $t$ [5, 13]. theorem 1 asserts that this algorithm is optimal if $H$ is a complete graph with at least 3 vertices, unless the SETH fails. A natural extension of this result would be to provide analogous tight bounds for non-complete targets $H$.

Egri, Marx, and Rzążewski [15] considered this problem in the setting of list homomorphisms. Let $H$ be a fixed graph. The input of the LHom($H$) problem consists of a graph $G$, whose every vertex is equipped with a list of vertices of the target $H$. We ask if $G$ has a homomorphism to $H$, respecting the lists. Egri et al. provided a full complexity classification for the case if $H$ is reflexive, i.e., every vertex has a loop. It is perhaps worth mentioning that a P / NP-complete dichotomy for LHom($H$) was first proved for reflexive graphs as well: If $H$ is a reflexive graph, then the LHom($H$) problem is polynomial time-solvable if $H$ is an interval graph, and NP-complete otherwise [17]. Egri et al. defined a new graph invariant $i^{*}(H)$, based on incomparable sets of vertices, and a new graph decomposition, and proved the following.

Theorem 2** (Egri, Marx, Rzążewski [15]).**

Let $H$ be a fixed non-interval reflexive graph with $i^{*}(H)=k$. Let $t$ be the treewidth of an instance graph $G$.

1. (a)

Assuming a tree decomposition of $G$ of width $t$ is given, the LHom($H$) problem can be solved in time $\mathcal{O}^{*}(k^{t})$. 2. (b)

There is no algorithm solving the LHom($H$) problem in time $\mathcal{O}^{*}\left((k-\varepsilon)^{t}\right)$ for any $\varepsilon>0$, unless the SETH fails.

In this paper we are interested in showing tight complexity bounds for the complexity of the non-list variant of the problem. Let us point out that despite the obvious similarity of Hom($H$) and LHom($H$) problems, they behave very differently when it comes to showing hardness results. Note that if $H^{\prime}$ is an induced subgraph of $H$, then any instance of LHom($H^{\prime}$) is also an instance of LHom($H$), where the vertices of $V(H)\setminus V(H^{\prime})$ do not appear in any list. Thus in order to prove hardness of LHom($H$), it is sufficient to find a “hard part” $H^{\prime}$ of $H$, and perform a reduction for the LHom($H^{\prime}$) problem. The complexity dichotomy for LHom($H$) was proven exactly along these lines [17, 18, 19]. Also the proof of theorem 2 (b) heavily uses the fact that we can work with some local subgraphs of $H$ and ignore the rest of vertices. In particular, all these proofs are purely combinatorial.

On the other hand, in the Hom($H$) problem, we need to capture the structure of the whole graph $H$, which is difficult using only combinatorial tools. This is why typical tools used in this area come from abstract algebra and algebraic graph theory.

For more information about graph homomorphisms we refer the reader to the comprehensive monograph by Hell and Nešetřil [28].

Our contribution

It is well known that in the study of graph homomorphisms the crucial role is played by the graphs that are cores, i.e., they do not have a homomorphism to any of its proper subgraphs. In particular, in order to provide a complete complexity classification of Hom($H$), it is sufficient to consider the case that $H$ is a connected core (we explain this in more detail in section 3). Also, the complexity dichotomy by Hell and Nešetřil [26] implies that Hom($H$) is polynomial-time solvable if $H$ is a graph on at most two vertices. So from now on let us assume that $H$ is a fixed core which is non-trivial, i.e., has at least three vertices.

We split the analysis into two cases, depending on the structure of $H$. First, in section 4.1, we consider targets $H$ that are projective (the definition of this class is rather technical, so we postpone it to section 2.2). We show that for projective cores the straightforward dynamic programming on a tree decomposition is optimal, assuming the SETH.

{restatable}

theoremprojective Let $H$ be a non-trivial projective core on $k$ vertices, and let $n$ and $t$ be, respectively, the number of vertices and the treewidth of an instance graph $G$.

1. (a)

Even if $H$ is given as a input, the Hom($H$) problem can be solved in time $\mathcal{O}(k^{4}+k^{t+1}\cdot n)$, assuming a tree decomposition of $G$ of width $t$ is given. 2. (b)

Even if $H$ is fixed, there is no algorithm solving the Hom($H$) problem in time $\mathcal{O}^{*}\left((k-\varepsilon)^{t}\right)$ for any $\varepsilon>0$, unless the SETH fails.

The proof brings together some tools and ideas from algebra and fine-grained complexity theory. The main technical ingredient is the construction of a so-called edge gadget, i.e., a graph $F$ with two specified vertices $u^{*}$ and $v^{*}$, such that:

1. (a)

for any distinct vertices $x,y$ of $H$, there is a homomorphism from $F$ to $H$, which maps $u^{*}$ to $x$ and $v^{*}$ to $y$, and 2. (b)

in any homomorphism from $F$ to $H$, the vertices $u^{*}$ and $v^{*}$ are mapped to distinct vertices of $H$.

Using this gadget, we can perform a simple and elegant reduction from $k$-Coloring. If $G$ is an instance of $k$-Coloring, we construct an instance $G^{*}$ of Hom($H$) by taking a copy of $G$ and replacing each edge $xy$ with a copy of the edge gadget, whose $u^{*}$-vertex is identified with $x$, and $v^{*}$-vertex is identified with $y$. By the properties of the edge gadget it is straightforward to observe that $G^{*}$ is $H$-colorable if and only if $G$ is $k$-colorable. Since the size of $F$ depends only on $H$, we observe that the treewidth of $G^{*}$ differs from the treewidth of $G$ by an additive constant, which is sufficient to obtain the desired lower bound.

Although the statement of section 1 might seem quite specific, it actually covers a large class of graphs. We say that a property $P$ holds for almost all graphs, if the probability that a graph chosen at random from the family of all graphs with vertex set $\{1,2,\ldots,n\}$ satisfies $P$ tends to 1 as $n\to\infty$. Hell and Nešetřil observed that almost all graphs are cores [27], see also [28, Corollary 3.28]. Moreover, Łuczak and Nešetřil proved that almost all graphs are projective [37]. From these two results, we can obtain that almost all graphs are projective cores. This, combined by section 1, implies the following.

Corollary 3**.**

*For almost all graphs $H$, the Hom($H$) problem on instance graphs with treewidth $t$ cannot be solved in time $\mathcal{O}^{*}\left((|H|-\varepsilon)^{t}\right)$ for any $\varepsilon>0$, unless the SETH fails. *

In section 4.2 we consider the case that $H$ is a non-projective core. First, we show that the approach that we used for projective cores cannot work in this case: it appears that one can construct the edge gadget for a core $H$ with the properties listed above if and only if $H$ is projective. What makes studying non-projective cores difficult is that we do not understand their structure well. In particular, we know that a graph $H=H_{1}\times H_{2}$, where $H_{1}$ and $H_{2}$ are non-trivial and $\times$ denotes the direct product of graphs (see section 2.2 for a formal definition), is non-projective, and by choosing $H_{1}$ and $H_{2}$ appropriately, we can ensure that $H$ is a core. However, we do not know whether there are any non-projective non-trivial connected cores that are indecomposable, i.e., they cannot be constructed using direct products. This problem was studied in a slightly more general setting by Larose and Tardif [34, Problem 2], and it remains wide open. We restate it here, only for restricted case that $H$ is a core, which is sufficient for our purpose.

{restatable}

conjectureconprojprime

Let $H$ be a connected non-trivial core. Then $H$ is projective if and only if it is indecomposable.

Since we do not know any counterexample to corollary 3, in the remainder we consider cores $H$ that are built using the direct product. If $H=H_{1}\times\ldots\times H_{m}$ and each $H_{i}$ is non-trivial and indecomposable, we call $H_{1}\times\ldots\times H_{m}$ a prime factorization of $H$. For such $H$ we show a lower complexity bound for Hom($H$), under an additional assumption that one of the factors $H_{i}$ of $H$ is truly projective.

The definition of truly projective graphs is rather technical and we present it in section 4.2. Graphs with such a property (actually, a slightly more restrictive one) were studied by Larose [32, Problems 1b. and 1b’.] in connection with some problems related to unique colorings, considered by Greenwell and Lovász [21]. Larose [32, 33] defined and investigated even more restricted class of graphs, called strongly projective (see section 5 for the definition). We know that every strongly projective graph is truly projective, and every truly projective graph is projective. Larose [32, 33] proved that all known projective graphs are in fact strongly projective. This raises a natural question whether projectivity and strong projectivity are in fact equivalent [32, 33]. Of course, an affirmative answer to this question would in particular mean that all projective cores are truly projective. Again, we state the problem in this weaker form, which is sufficient for our application.

{restatable}

conjectureconjstrongly Every projective core is truly projective.

Actually, if we assume both corollary 3 and corollary 3, we are able to provide a full complexity classification for the Hom($H$) problem, parameterized by the treewidth of the input graph.

{restatable}

theoremthmeverything Assume that corollary 3 and corollary 3 hold. Let $H$ be a non-trivial connected core with prime factorization $H_{1}\times\ldots\times H_{m}$, and define $k:=\max_{i\in[m]}|H_{i}|$. Let $n$ and $t$ be, respectively, the number of vertices and the treewidth of an instance graph $G$.

1. (a)

Even if $H$ is given as an input, the Hom($H$) problem can be solved in time $\mathcal{O}(|H|^{4}+k^{t+1}\cdot n)$, assuming a tree decomposition of $G$ of width $t$ is given. 2. (b)

Even if $H$ is fixed, there is no algorithm solving the Hom($H$) problem in time $\mathcal{O}^{*}\left((k-\varepsilon)^{t}\right)$ for any $\varepsilon>0$, unless the SETH fails.

Let us point out that despite some work on both conjectures [34, 32, 33], we know no graph $H$ for which the bounds from corollary 3 do not hold.

2 Notation and preliminaries

For $n\in\mathbb{N}$, we denote the set $\{1,2,\ldots,n\}$ by $[n]$. All graphs considered in this paper are finite, undirected and do not contain parallel edges. For a graph $G$, by $V(G)$ and $E(G)$ we denote the set of vertices and the set of edges of $G$, respectively, and we write $|G|$ for the number of vertices of $G$. Let $K_{1}^{*}$ be the single-vertex graph with a loop. A graph is ramified if it has no two distinct vertices $u$ and $v$ such that the open neighborhood of $u$ is contained in the open neighborhood of $v$. An odd girth of a graph $G$, denoted by $\operatorname{og}(G)$, is the length of a shortest odd cycle in $G$. For a graph $G$, denote by $\omega(G)$ and $\chi(G)$, respectively, the size of the largest clique contained in $G$ and the chromatic number of $G$.

A tree decomposition of a graph $G$ is a pair $\left(\mathcal{T},\{X_{a}\}_{a\in V(\mathcal{T})}\right)$, in which $\mathcal{T}$ is a tree, whose vertices are called nodes and $\{X_{a}\}_{a\in V(\mathcal{T})}$ is the family of subsets (called bags) of $V(G)$, such that

every $v\in V(G)$ belongs to at least one bag $X_{a}$, 2. 2.

for every $uv\in E(G)$ there is at least one bag $X_{a}$ such that $u,v\in X_{a}$, 3. 3.

for every $v\in V(G)$ the set $\mathcal{T}_{v}:=\{a\in V(\mathcal{T})\leavevmode\nobreak\ |\leavevmode\nobreak\ v\in X_{a}\}$ induces a connected subgraph of $\mathcal{T}$.

The width of a tree decomposition $\left(\mathcal{T},\{X_{a}\}_{a\in V(\mathcal{T})}\right)$ is the number $\max_{a\in V(\mathcal{T})}|X_{a}|-1$. The minimum possible width of a tree decomposition of $G$ is called the treewidth of $G$ and denoted by ${\operatorname{tw}(G)}$. In particular, if $\mathcal{T}$ is a path, then a tree decomposition $\left(\mathcal{T},\{X_{a}\}_{a\in V(\mathcal{T})}\right)$ is called a path decomposition. The pathwidth of $G$, denoted by ${\operatorname{pw}(G)}$, is the minimum possible width of a path decomposition of $G$. Clearly for every graph $G$ it holds that ${\operatorname{tw}(G)}\leqslant{\operatorname{pw}(G)}$.

2.1 Graph homomorphisms and cores

For graphs $G$ and $H$, a function $f:V(G)\to V(H)$ is a homomorphism if it preserves edges, i.e., for every $uv\in E(G)$ it holds that $f(u)f(v)\in E(H)$ (see fig. 1). If $G$ admits a homomorphism to $H$, we denote this fact by $G\to H$ and we write $f:G\to H$ if $f$ is a homomorphism from $G$ to $H$. If there is no homomorphism from $G$ to $H$, we write $G\not\to H$. Graphs $G$ and $H$ are homomorphically equivalent if $G\to H$ and $H\to G$, and incomparable if $G\not\to H$ and $H\not\to G$. Observe that homomorphic equivalence is an equivalence relation on the class of all graphs. An endomorphism of $G$ is any homomorphism $f\colon G\to G$.

A graph $G$ is a core if $G\not\to H$ for every proper subgraph $H$ of $G$. Equivalently, we can say $G$ is a core if and only if every endomorphism of $G$ is an automorphism (i.e., an isomorphism from $G$ to $G$). Note that a core is always ramified. If $H$ is a subgraph of $G$ such that $G\to H$ and $H$ is a core, we say that $H$ is a core of $G$. Notice that if $H$ is a subgraph of $G$, then it always holds that $H\to G$, so every graph is homomorphically equivalent to its core. Moreover, if $H$ is a core of $G$, then $H$ is always an induced subgraph of $G$, because every endomorphism $f:G\to H$ restricted to $H$ must be an automorphism. It was observed by Hell and Nešetřil that every graph has a unique core (up to an isomorphism) [27]. Note that if $f:G\to H$ is a homomorphism from $G$ to its core $H$, then it must be surjective.

We say that a core is trivial if it is isomorphic to $K_{1}$, $K_{1}^{*}$, or $K_{2}$. It is easy to observe that these three graphs are the only cores with fewer than 3 vertices. In general, finding a core of a given graph is computationally hard; in particular, deciding if a graph is a core is coNP-complete [27]. However, the graphs whose cores are trivial are simple to describe.

Observation 4**.**

Let $G$ be a graph, whose core $H$ is trivial.

1. (a)

$H\simeq K_{1}$* if and only if $\chi(G)=1$, i.e., $G$ has no edges,* 2. (b)

$H\simeq K_{2}$* if and only if $\chi(G)=2$, i.e., $G$ is bipartite and has at least one edge,* 3. (c)

$H\simeq K_{1}^{*}$* if and only if $G$ has a vertex with a loop. $\square$*

In particular, there are no non-trivial cores with loops. The following conditions are necessary for $G$ to have a homomorphism into $H$.

Observation 5** ([28]).**

*Assume that $G\to H$ and $G$ and $H$ have no loops. Then $\omega(G)\leqslant\omega(H)$, $\chi(G)\leqslant\chi(H)$, and $\operatorname{og}(G)\geqslant\operatorname{og}(H)$. $\square$ *

We denote by $H_{1}+\ldots+H_{m}$ a disconnected graph with connected components $H_{1},\dots,H_{m}$. Observe that if $f$ is a homomorphism from $G=G_{1}+\ldots+G_{\ell}$ to $H=H_{1}+\ldots+H_{m}$, then it maps every connected component of $G$ into some connected component of $H$. Also note that a graph does not have to be connected to be a core, in particular the following characterization follows directly from the definition of a core.

Observation 6**.**

*A disconnected graph $H$ is a core if and only if its connected components are pairwise incomparable cores. $\square$ *

An example of a pair of incomparable cores is shown in fig. 2: it is the Grötzsch graph, denoted by $G_{G}$, and the clique $K_{3}$. Clearly, $\operatorname{og}(G_{G})>\operatorname{og}(K_{3})$ and $\chi(G_{G})>\chi(K_{3})$, so by 5, they are incomparable. Therefore, by 6, the graph $G_{G}+K_{3}$ is a core.

Finally, let us observe that we can construct arbitrarily large families of pairwise incomparable cores. Let us start the construction with an arbitrary non-trivial core $H_{0}$. Now suppose we have constructed pairwise incomparable cores $H_{0},H_{1},\ldots,H_{k},$ and we want to construct $H_{k+1}$. Let $\ell=\max_{i\in\{0,\ldots,k\}}\operatorname{og}(H_{i})$, $r=\max_{i\in\{0,\ldots,k\}}\chi(H_{i})$. By the classic result of Erdős [16], there is a graph $H$ with $\operatorname{og}(H)>\ell$ and $\chi(H)>r$. We set $H_{k+1}$ to be the core of $H$. Observe that $\operatorname{og}(H_{k+1})=\operatorname{og}(H)>\ell$ and $\chi(H_{k+1})=\chi(H)>r$, so, by 5, we have that for every $i\in\{0,\ldots,k\}$ the core $H_{k+1}$ is incomparable with $H_{i}$.

2.2 Graph products and projectivity

Define the direct product of graphs $H_{1}$ and $H_{2}$, denoted by $H_{1}\times H_{2}$, as follows:

[TABLE]

If $H=H_{1}\times H_{2}$, then $H_{1}\times H_{2}$ is a factorization of $H$, and $H_{1}$ and $H_{2}$ are its factors. Clearly, the binary operation $\times$ is commutative, so will identify $H_{1}\times H_{2}$ and $H_{2}\times H_{1}$. Since $\times$ is also associative, we can extend the definition for more than two factors:

[TABLE]

Moreover, in the next sections, we will sometimes consider products of graphs, that are products themselves. Formally, the vertices of such graphs are tuples of tuples. If it does not lead to confusion, for $\bar{x}:=(x_{1},\dots,x_{k_{1}})$ and $\bar{y}:=(y_{1},\dots,y_{k_{2}})$, we will treat tuples $(\bar{x},\bar{y}),$ $(x_{1},\dots,x_{k_{1}},y_{1},\dots,y_{k_{2}})$, $(\bar{x},y_{1},\dots,y_{k_{2}})$, and $(x_{1},\dots,x_{k_{1}},\bar{y})$ as equivalent. This notation is generalized to more factors in a natural way. We denote by $H^{m}$ the product of $m$ copies of $H$.

The direct product appears in the literature under different names: tensor product, cardinal product, Kronecker product, relational product. It is also called categorical product, because it is the product in the category of graphs (see [23, 39] for details).

Note that if $H_{1}\times H_{2}$ has at least one edge, then $H_{1}\times H_{2}\simeq H_{1}$ if and only if $H_{2}\simeq K_{1}^{*}$. We say that a graph $H$ is directly indecomposable (or indecomposable for short) if the fact that $H=H_{1}\times H_{2}$ implies that either $H_{1}\simeq K_{1}^{*}$ or $H_{2}\simeq K_{1}^{*}$. A graph that is not indecomposable, is decomposable. A factorization, where each factor is directly indecomposable and not isomorphic to $K_{1}^{*}$, is called a prime factorization. Clearly, $K_{1}^{*}$ does not have a prime factorization.

The following property will be very useful (see also Theorem 8.17 in [23]).

Theorem 7** (McKenzie [38]).**

*Any connected non-bipartite graph with more than one vertex has a unique prime factorization into directly indecomposable factors (with possible loops). *

Let $H_{1}\times\ldots\times H_{m}$ be some factorization of $H$ (not necessary prime) and let $i\in[m]$. A function $\pi_{i}:V(H)\to V(H_{i})$ such that for every $(x_{1},\dots,x_{m})\in V(H)$ it holds that $\pi_{i}(x_{1},\dots,x_{m})=x_{i}$ is a projection on the $i$-th coordinate. It follows from the definition of the direct product that every projection $\pi_{i}$ is a homomorphism from $H$ to $H_{i}$.

Below we summarize some basic properties of direct products.

Observation 8**.**

Let $H$ be a graph on $k$ vertices. Then

1. (a)

$H\times K_{1}$* consists of $k$ isolated vertices, in particular its core is $K_{1}$,* 2. (b)

if $H$ has at least one edge, then the core of $H\times K_{2}$ is $K_{2}$, 3. (c)

the graph $H^{m}$ contains a subgraph isomorphic to $H$, which is induced by the set $\{(x,\dots,x)\leavevmode\nobreak\ |\leavevmode\nobreak\ x\in V(H)\}$; in particular, if $m\geqslant 2$, then $H^{m}$ is never a core, 4. (d)

if $H=H_{1}\times\ldots\times H_{m}$ and $H_{1},H_{2},\ldots,H_{m}$ are connected, then $H$ is connected if and only if at most one $H_{i}$ is bipartite, 5. (e)

if $H=H_{1}\times\ldots\times H_{m}$, then for every $G$ it holds that $G\to H$ if and only if $G\to H_{i}$ for all $i\in[m]$.

Proof.

Items (a), (b), (c) are straightforward to observe. Item (d) follows from a result of Weichsel [51], see also [23, Corollary 5.10]. To prove (e), consider a homomorphism $f:G\to H$. Clearly, $H\to H_{i}$ for every $i\in[m]$ because each projection $\pi_{i}:H\to H_{i}$ is a homomorphism. So $\pi_{i}\circ f$ is a homomorphism from $G$ to $H_{i}$. On the other hand, if we have some $f_{i}:G\to H_{i}$ for every $i\in[m]$, then we can define a homomorphism $f:G\to H$ by $f(x):=(f_{1}(x),\dots,f_{m}(x))$.

A homomorphism $f:H^{m}\to H$ is idempotent, if for every $x\in V(H)$ it holds that $f(x,x,\dots,x)=x$. One of the main characters of the paper is the class of projective graphs, considered e.g. in [32, 33, 34]. A graph $H$ is projective (or idempotent trivial), if for every $m\geqslant 2$, every idempotent homomorphism from $H^{m}$ to $H$ is a projection.

Observation 9**.**

*If $H$ is a projective core and $f:H^{m}\to H$ is a homomorphism, then $f\equiv g\circ\pi_{i}$ for some $i\in[m]$ and some automorphism $g$ of $H$. *

Proof.

If $f$ is idempotent, then it is a projection and we are done. Assume $f$ is not idempotent and define $g:V(H)\to V(H)$ by $g(x)=f(x,\dots,x)$. The function $g$ is an endomorphism of $H$ and $H$ is a core, so $g$ is in fact an automorphism of $H$. Observe that $g^{-1}\circ f$ is an idempotent homomorphism, so it is equal to $\pi_{i}$ for some $i\in[m]$, because $H$ is projective. From this we get that $f\equiv g\circ\pi_{i}$.

It is known that projective graphs are always connected [34]. Observe that the definition of projective graphs does not imply that their recognition is decidable. However, an algorithm to recognize these graphs follows from the following, useful characterization.

Theorem 10** (Larose, Tardif [34]).**

*A connected graph $H$ with at least three vertices is projective if and only if every idempotent homomorphism from $H^{2}$ to $H$ is a projection. *

Recall from the introduction that almost all graphs are projective cores [28, 37]. It appears that the properties of projectivity and being a core are independent. In particular, the graph in fig. 3 is not a core, as it can be mapped to a triangle. However, Larose [32] proved that all non-bipartite, connected, ramified graphs which do not contain $C_{4}$ as a (non-necessarily induced) subgraph, are projective (this will be discussed in more detail in section 5, see theorem 24). On the other hand, there are also non-projective cores, an example is $G_{G}\times K_{3}$, see fig. 2. We discuss such graphs in detail in section 4.2.

3 Complexity of finding graph homomorphisms

Note that if two graphs $H_{1}$ and $H_{2}$ are homomorphically equivalent, then the Hom($H_{1}$) and Hom($H_{2}$) problems are also equivalent. So in particular, because every graph is homomorphically equivalent to its core, we may restrict our attention to graphs $H$ which are cores. Also, recall from 4 that Hom($H$) can be solved in polynomial time if $H$ is isomorphic to $K_{1}^{*}$, $K_{1}$, or $K_{2}$. So we will be interested only in non-trivial cores $H$. In particular, we will assume that $H$ is non-bipartite and has no loops.

We are interested in understanding the complexity bound of the Hom($H$) problem, parameterized by the treewidth of the input graph. The dynamic programming approach (see Bodlaender et al. [5]) gives us the following upper bound.

Theorem 11** (Bodlaender et al. [5]).**

*Let $H$ be a graph on $k$ vertices. Even if $H$ is a part of the input, the Hom($H$) problem can be solved in time $\mathcal{O}(k^{t+1}\cdot n)$, assuming a tree decomposition of width $t$ of the instance graph on $n$ vertices is given. *

By theorem 1, this bound is tight (up to the polynomial factor) if $H$ is a complete graph with at least three vertices, unless the SETH fails. We are interested in extending this result for other graphs $H$.

First, let us observe that there are cores, for which the bound from theorem 11 can be improved. Indeed, let $H$ be a decomposable core, isomorphic to $H_{1}\times\ldots\times H_{m}$ (see discussion in section 4.2 for more about cores that are products.). Recall from 8 (e) that for every graph $G$ it holds that

[TABLE]

So, given an instance $G$ of Hom($H$), we can call the algorithm from theorem 11 to solve Hom($H_{i}$) for each $i\in[m]$ and return a positive answer if and only if we get a positive answer in each of the calls. Since Hammack and Imrich [24] presented an algorithm for finding the prime factorization of $H$ in time $\mathcal{O}(|H|^{4})$, we obtain the following result.

Theorem 12**.**

*Let $H$ be a core with prime factorization $H_{1}\times\ldots\times H_{m}$. Even if $H$ is given as a part of the input, the Hom($H$) problem can be solved in time $\mathcal{O}\left(|H|^{4}+\max_{j\in[m]}|H_{j}|^{t+1}\cdot n\right)$, assuming a tree decomposition of width $t$ of the instance graph with $n$ vertices is given. $\square$ *

Let us conclude this section with two lemmas, which justify why we restrict our attention to connected cores. Assume $H=H_{1}+\ldots+H_{m}$ is a disconnected core. The first lemma shows how to solve Hom($H$) using an algorithm for finding homomorphisms to connected target graphs.

Lemma 13**.**

*Consider a disconnected core $H=H_{1}+\ldots+H_{m}$. Assume that for every $i\in[m]$ the Hom($H_{i}$) problem can be solved in time $\mathcal{O}(|H_{i}|^{4}+n^{d}\cdot c(H_{i})^{t})$ for an instance $G$ with $n$ vertices, given along with its tree decomposition of width $t$, where $c$ is some function and $d$ is a constant. Then, even in $H$ is given as an input, the Hom($H$) problem can be solved in time $\mathcal{O}\left(|H|^{4}+n^{d}\cdot\max_{i\in[m]}c(H_{i})^{t}\cdot|H|\right)$. *

Proof.

First, observe that if $G$ is disconnected, say $G=G_{1}+\ldots+G_{\ell}$, then $G\to H$ if and only if $G_{i}\to H$ for every $i\in[\ell]$. Also, any tree decomposition of $G$ can be easily transformed to a tree decomposition of $G_{i}$ of at most the same width. It means that if the instance graph is disconnected, we can just consider the problem separately for each of its connected components. So we assume that $G$ is connected. Then $G\to H$ if and only if $G\to H_{i}$ for some $i\in[m]$. We find the connected components of $H$ in time $\mathcal{O}(|H|^{2})$ and then solve Hom($H_{i}$) for each $i\in[m]$ (for the same instance $G$) in time $\mathcal{O}(|H_{i}|^{4}+n^{d}\cdot c(H_{i})^{t})$. We return a positive answer for Hom($H$) if and only if we get a positive answer for at least one $i\in[m]$. The total complexity of this algorithm is $\mathcal{O}\left(|H|^{2}+|H_{1}|^{4}+\ldots+|H_{m}|^{4}+n^{d}\cdot(c(H_{1})^{t}+\ldots+c(H_{m})^{t})\right)=\mathcal{O}\left(|H|^{4}+n^{d}\cdot\max_{i\in[m]}c(H_{i})^{t}\cdot|H|\right),$ as $|H_{1}|^{4}+\ldots+|H_{m}|^{4}\leqslant(|H_{1}|+\ldots+|H_{m}|)^{4}=|H|^{4}$.

The second lemma shows that even if we assume $H$ to be fixed, we cannot solve Hom($H$) faster than we solve Hom($H_{i}$), for each connected component $H_{i}$ of $H$. We state it in a stronger version, parameterized by the pathwidth of an instance graph.

Lemma 14**.**

*Let $H=H_{1}+\ldots+H_{m}$ be a fixed, disconnected core. Assume that the Hom($H$) problem can be solved in time $\mathcal{O}^{*}(\alpha^{\operatorname{pw}(G)})$ for an instance $G$ given along with its optimal path decomposition. Then for every $i\in[m]$ the Hom($H_{i}$) problem can be solved in time $\mathcal{O}^{*}(\alpha^{{\operatorname{pw}(G)}})$. *

Proof.

Again, we may restrict our attention only to connected instances, as otherwise we can solve the problem separately for each instance. Consider a connected instance $G$ of Hom($H_{i}$) on $n$ vertices and pathwidth $t$. Let $V(H_{i})=\{z_{1},\dots,z_{k}\}$ and let $u$ be some fixed vertex of $G$. We construct an instance $G^{*}$ of Hom($H$) as follows. We take a copy $G^{\prime}$ of $G$ and a copy $\widetilde{H}_{i}^{k}$ of $H_{i}^{k}$, and identify the vertex corresponding to $u$ in $G^{\prime}$ with the vertex corresponding to $(z_{1},\dots,z_{k})$ in $\widetilde{H}_{i}^{k}$. Denote this vertex of $G^{*}$ by $\bar{z}$. Observe that $H_{i}$ a connected, non-trivial core, so 8 (d) implies that $\widetilde{H}_{i}^{k}$ is connected. Since $G$ is also connected, $G^{*}$ must be connected.

We claim that $G\to H_{i}$ if and only if $G^{*}\to H$. Indeed, if $f:G\to H_{i}$, then there exists $j\in[k]$ such that $f(u)=z_{j}$, so we can define a homomorphism $g:G^{*}\to H_{i}$ (which is also a homomorphism from $G^{*}$ to $H$) by

[TABLE]

Clearly, both $f$ and $\pi_{j}$ are homomorphisms. Recall that $\bar{z}$ is a cutvertex in $G^{*}$ obtained by identifying $u$ from $G^{\prime}$ and $(z_{1},\dots,z_{k})$ from $\widetilde{H}_{i}^{k}$. Furthermore, we have $g(\bar{z})=f(u)=\pi_{j}(z_{1},\dots,z_{k})=z_{j}$, so $g$ is a homomorphism from $G^{*}$ to $H$.

Conversely, if we have $g:G^{*}\to H$, we know that $g$ maps $G^{*}$ to a connected component $H_{j}$ of $H$, for some $j\in[m]$, because $G^{*}$ is connected. But $G^{*}$ contains an induced copy $\widetilde{H}_{i}^{k}$ of $H_{i}^{k}$, so also an induced copy of $H_{i}$, say $\widetilde{H}_{i}$ (recall 8 (c)). So $g|_{V(\widetilde{H}_{i})}$ is in fact a homomorphism from $H_{i}$ to $H_{j}$. Recall from 5 that since $H_{1}+\ldots+H_{m}$ is a core, its connected components are pairwise incomparable cores – so $j$ must be equal to $i$. It means that $g|_{V(G^{\prime})}$ is a homomorphism from $G^{\prime}$ to $H_{i}$, so we conclude that $G\to H_{i}$.

Note that the number of vertices of $G^{*}$ is $n+|H_{i}^{k}|-1\leqslant|H_{i}^{k}|\cdot n$. Now let $\left(\mathcal{T},\{X_{a}\}_{a\in V(\mathcal{T})}\right)$ be a path decomposition of $G$ of width $t$, and let $b$ be a node of $\mathcal{T}$, such that $u\in X_{b}$. Let $\mathcal{T}^{*}$ be the path obtained from $\mathcal{T}$ by inserting a new node $b^{\prime}$ as the direct successor of $b$. Define $X_{b^{\prime}}:=X_{b}\cup V(H_{i}^{k})$. Clearly, $\left(\mathcal{T}^{*},\{X_{a}\}_{a\in V(\mathcal{T}^{*})}\right)$ is a path decomposition of $G^{*}$. This means that ${\operatorname{pw}(G^{*})}\leqslant t+|H_{i}^{k}|$. The graph $H_{i}$ is fixed, so the number of vertices of $H_{i}^{k}$ is a constant. By our assumption we can decide if $G^{*}\to H$ in time $\alpha^{\operatorname{pw}(G^{*})}\cdot c\cdot|G^{*}|^{d}$, so we can decide if $G\to H_{i}$ in time $\alpha^{\operatorname{pw}(G^{*})}\cdot c\cdot(|H_{i}^{k}|n)^{d}\leqslant\alpha^{t}\alpha^{|H_{i}^{k}|}\cdot c\cdot|H_{i}^{k}|^{d}n^{d}=\alpha^{t}\cdot c^{\prime}\cdot n^{d}$, where $c^{\prime}=c\cdot\alpha^{|H_{i}^{k}|}\cdot|H_{i}^{k}|^{d}$.

Let us point out that the assumptions in lemma 13 (that $H$ is given as an input) and lemma 14 (that $H$ is fixed) correspond, respectively, to the assumptions in statements (a) and (b) of section 1 and corollary 3.

4 Lower bounds

In this section we will investigate the lower bounds for the complexity of Hom($H$). The section is split into two main parts. In section 4.1 we consider projective cores. Then, in section 4.2, we consider non-projective cores.

4.1 Projective cores

The main result of this section is section 1. \projective*

Observe that section 1 (a) follows from theorem 11, so we need to show the hardness counterpart, i.e., the statement (b). A crucial building block in our reduction will be the graph called the edge gadget, whose construction is described in the following lemma.

Lemma 15**.**

For every non-trivial projective core $H$, there exists a graph $F$ with two specified vertices $u^{*}$ and $v^{*}$, satisfying the following:

1. (a)

for every $x,y\in V(H)$ such that $x\neq y$, there exists a homomorphism $f:F\to H$ such that $f(u^{*})=x$ and $f(v^{*})=y$, 2. (b)

for every $f:F\to H$ it holds that $f(u^{*})\neq f(v^{*})$.

Proof.

Let $V(H)=\{z_{1},\dots,z_{k}\}$. For $i\in[k]$ denote by $z_{i}^{k-1}$ the $(k-1)$-tuple $(z_{i},\dots,z_{i})$ and by $\overline{z_{i}}$ the $(k-1)$-tuple $(z_{1},\dots,z_{i-1},z_{i+1},\dots z_{k})$. We claim that $F:=H^{(k-1)k}$ and vertices

[TABLE]

satisfy the statement of the lemma. Note that the vertices of $F$ are $k(k-1)$-tuples.

To see that (a) holds, observe that if $x$ and $y$ are distinct vertices from $V(H)$, then there always exists $i\in[k(k-1)]$ such that $\pi_{i}(u^{*})=x$ and $\pi_{i}(v^{*})=y$. This means that $\pi_{i}$ is a homomorphism from $F=H^{k(k-1)}$ to $H$ satisfying $\pi_{i}(u^{*})=x$ and $\pi_{i}(v^{*})=y$.

To prove (b), recall that since $H$ is projective, by 9, the homomorphism $f$ is a composition of some automorphism $g$ of $H$ and $\pi_{i}$ for some $i\in[k(k-1)]$. Observe that $u^{*}$ and $v^{*}$ are defined in a way such that $\pi_{j}(u^{*})\neq\pi_{j}(v^{*})$ for every $j\in[k(k-1)]$. As $g$ is an automorphism, it is injective, which gives us $f(u^{*})=g(\pi_{i}(u^{*}))\neq g(\pi_{i}(v^{*}))=f(v^{*})$.

Finally, we are ready to prove section 1 (b). The high-level idea is to start with an instance of $k$-Coloring, where $k=|H|$, and replace each edge by the gadget constructed in lemma 15. Then the hardness will follow from theorem 1. Actually, Lokshtanov, Marx, Saurabh [36] proved the following, slightly stronger version of theorem 1.

Theorem 1’** (Lokshtanov, Marx, Saurabh [36]).**

*Let $k\geqslant 3$ be a fixed integer. Assuming the SETH, the $k$-Coloring problem on a graph $G$ cannot be solved in time $\mathcal{O}^{*}\left((k-\varepsilon)^{{\operatorname{pw}(G)}}\right)$ for any $\varepsilon>0$. *

This allows us to prove a slightly stronger version of section 1 (b), where we consider the problem parameterized by the pathwidth of the instance graph.

Theorem 3’ (b)****.

*Let $H$ be a fixed non-trivial projective core on $k$ vertices. There is no algorithm solving the Hom($H$) problem for instance graph $G$ in time $\mathcal{O}^{*}\left((k-\varepsilon)^{{\operatorname{pw}(G)}}\right)$ for any $\varepsilon>0$, unless the SETH fails. *

Proof of 3’ (b).

Note that since $H$ is non-trivial, we have $k\geqslant 3$. Recall that since $H$ is projective, it is also connected [34]. We reduce from $k$-Coloring, let $G$ be an instance with $n$ vertices and pathwidth $t$. We construct an instance $G^{*}$ of Hom($H$) as follows. First, for every $z\in V(G)$ we introduce a vertex $z^{\prime}$ of $V(G^{*})$. Let $V^{\prime}$ denote the set of these vertices. Now, for every edge $xy$ of $G$, we introduce to $G^{*}$ a copy of the edge gadget, constructed in lemma 15, and denote it by $F_{xy}$. We identify the vertices $u^{*}$ and $v^{*}$ of $F_{xy}$ with vertices $x^{\prime}$ and $y^{\prime}$, respectively. This completes the construction of $G^{*}$.

We claim that $G$ is $k$-colorable if and only if $G^{*}\to H$. Indeed, let $\varphi$ be a $k$-coloring of $G$. For simplicity of notation, we label the colors used by $\varphi$ in the same way as the vertices of $H$, i.e., $z_{1},z_{2},\dots,z_{k}$. Define $g:V^{\prime}\to V(H)$ by setting $g(v^{\prime}):=\varphi(v^{\prime})$ Now consider an edge $xy$ of $G$ and the edge gadget $F_{xy}$. Since $c$ is a proper coloring, we have $g(x^{\prime})\neq g(y^{\prime})$. So by lemma 15 (a), we can find a homomorphism $f_{xy}:F_{xy}\to H$, such that $f_{xy}(x^{\prime})=g(x^{\prime})$ and $f_{xy}(y^{\prime})=g(y^{\prime})$. Repeating this for every edge gadget, we can extend $g$ to a homomorphism from $G^{*}$ to $H$.

Conversely, from lemma 15 (b), we know that for any $f\colon G^{*}\to H$ and every edge $xy$ of $G$ it holds that $f(x^{\prime})\neq f(y^{\prime})$, so any homomorphism from $G^{*}$ to $H$ induces a $k$-coloring of $G$.

The number of vertices of $G^{*}$ is at most $|F|n^{2}$. Now let $\mathcal{T}$ be a path decomposition of $G$ of width $t$, denote its consecutive bags by $X_{1},X_{2},\ldots,X_{m}$. Let us extend it to a path decomposition of $G^{*}$. For each edge $xy$ of $G$ there exists $b\in[m]$ such that $x,y\in X_{b}$. We introduce a bag $X_{b^{\prime}}:=X_{b}\cup V(F_{xy})$ as a direct successor of $X_{b}$. It is straightforward to observe that by repeating this step for every edge of $G$, we obtain a path decomposition of $G^{*}$ of width at most $t+|F|$. Recall that $H$ is fixed, so $|F|$ is a constant. So if we could decide if $G^{*}\to H$ in time $(k-\varepsilon)^{\operatorname{pw}(G^{*})}\cdot c\cdot|G^{*}|^{d}\leqslant(k-\varepsilon)^{t+|F|}\cdot c\cdot|F|^{d}\cdot n^{2d}$, where $c$ and $d$ are some constants, then we would be able to decide if $G$ is $k$-colorable in time $(k-\varepsilon)^{t}\cdot c^{\prime}\cdot n^{d^{\prime}}$ for constants $c^{\prime}=c\cdot(k-\varepsilon)^{|F|}\cdot|F|^{d}$ and $d^{\prime}=2d$. By 1’, this contradicts the SETH.

4.2 Non-projective cores

Now we will focus on non-trivial connected cores, which are additionally non-projective, i.e., they do not satisfy the assumptions of section 1. First, let us argue that the approach from section 4.1 cannot work in this case. In particular, we will show that an edge gadget with properties listed in lemma 15 cannot be constructed for non-projective graphs $H$.

We will need the definition of constructible sets, see Larose and Tardif [34]. For a graph $H$, a set $C\subseteq V(H)$ is constructible if there exists a graph $K$, an $(\ell+1)$-tuple of vertices $x_{0},\dots,x_{\ell}\in V(K)$ and an $\ell$-tuple of vertices $y_{1},\dots,y_{\ell}\in V(H)$ such that

[TABLE]

We can think of $C$ as the set of colors that might appear on the vertex $x_{0}$, when we precolor each $x_{i}$ with the color $y_{i}$ and try to extend this partial mapping to a homomorphism to $H$. The tuple $(K,x_{0},\ldots,x_{\ell},y_{1},\ldots,y_{\ell})$ is called a construction of $C$.

It appears that the notion of constructible sets is closely related to projectivity.

Theorem 16** (Larose, Tardif [34]).**

*A graph $H$ on at least three vertices is projective if and only if every subset of its vertices is constructible. *

Now we show that lemma 15 cannot work for non-projective graphs $H$.

Proposition 17**.**

*Let $H$ be a fixed non-trivial connected core. Then an edge gadget $F$ with properties listed in lemma 15 exists if and only if $H$ is projective. *

Proof.

The ‘if’ statement follows from lemma 15. Let $k:=|H|$ and suppose that there exists a graph $F$ with properties given in lemma 15. Consider a set $C\subseteq V(H)$ and define $\ell:=|C|$. Let $\{y_{1},\dots,y_{k-\ell}\}$ be the complement of $C$ in $V(H)$. Take $k-\ell$ copies of $F$, say $F_{1},\dots,F_{k-\ell}$ and denote the vertices $u^{*}$ and $v^{*}$ of the $i$-th copy $F_{i}$ by $u^{*}_{i}$ and $v^{*}_{i}$, respectively. Identify the vertices $u^{*}_{i}$ of all these copies, denote the obtained vertex by $u^{*}$, and the obtained graph by $K$. Now set $x_{0}:=u^{*}$ and for each $i\in[k-\ell]$ set $x_{i}:=v^{*}_{i}$.

It is easy to verify that this is a construction of the set $C$. Indeed, observe that if $x\in C$, then, from lemma 15 (a), for each copy $F_{i}$ there exists a homomorphism $f_{i}:F_{i}\to H$ such that $f_{i}(v^{*}_{i})=f_{i}(x_{i})=y_{i}$ and $f_{i}(u^{*})=f_{i}(x_{0})=x$. Combining these homomorphisms yields a homomorphism $f:K\to H$. On the other hand, if $x\not\in C$, then $x=y_{i}$ for some $i\in[k-\ell]$. But from lemma 15 (b) we know that for every homomorphism $f\colon F_{i}\to H$ it holds that $x=y_{i}=f(v^{*}_{i})\neq f(u^{*})=f(x_{0})$, so $x_{0}$ cannot be mapped to $x$ by any extension to a homomorphism from $K$ to $H$.

Observe that if $H$ is projective, then it must be indecomposable. Indeed, assume that for some non-trivial $H$ it holds that $H=H_{1}\times H_{2}$, $H\not\simeq K_{1}^{*}$ and $H_{2}\not\simeq K_{1}^{*}$. Consider a homomorphism $f:(H_{1}\times H_{2})^{2}\to H_{1}\times H_{2}$, defined as $f((x,y),(x^{\prime},y^{\prime}))=(x,y^{\prime})$. Note that it is idempotent, but not a projection, so $H$ is not projective.

In the light of the observation above, it is natural to ask whether indecomposability implies projectivity. This problem was already stated e.g. by Larose and Tardif [34, Problem 2] and, to the best of our knowledge, no significant progress in this direction was made. Let us recall it here.

\conprojprime

Since we know no connected non-trivial non-projective cores that are indecomposable, in the remainder of the section we will assume that $H$ is a decomposable, non-trivial connected core. By theorem 7 we know that $H$ has a unique prime factorization $H_{1}\times\ldots\times H_{m}$ for some $m\geqslant 2$. To simplify the notation, for any given homomorphism $f:G\to H_{1}\times\ldots\times H_{m}$ and $i\in[m]$, we define $f_{i}\equiv\pi_{i}\circ f$. Then for each vertex $x$ of $G$ it holds that

[TABLE]

and $f_{i}$ is a homomorphism from $G$ to $H_{i}$.

The following observation follows from 8.

Observation 18**.**

*Let $H$ be a connected, non-trivial core with factorization $H=H_{1}\times\ldots\times H_{m}$, such that $H_{i}\not\simeq K_{1}^{*}$ for all $i\in[m]$. Then for $i\in[m]$ the graph $H_{i}$ is a connected non-trivial core, incomparable with $H_{j}$ for $j\in[m]\setminus\{i\}$. $\square$ *

Now let us consider the complexity of Hom($H$), where $H$ has a prime factorization $H_{1}\times H_{2}\times\ldots H_{m}$ for $m\geqslant 2$. By theorem 12, the problem can be solved in time $\mathcal{O}\left(|H|^{4}+\max_{j\in[m]}|H_{j}|^{t+1}\cdot n\right)$, where $n$ and $t$ are, respectively, the number of vertices and the width of a given tree decomposition of the input graph. We believe that this bound is tight, and we prove a matching lower bound, up to the polynomial factor, under some additional assumption.

We say that a graph $H$ is truly projective if it has at least three vertices and for every $s\geqslant 2$ and every connected core $W$ incomparable with $H$, it holds that if $f:H^{s}\times W\to H$ satisfies $f(x,x,\dots,x,y)=x$ for all $x\in V(H),y\in V(W)$, then $f$ is a projection.

It is easy to verify that truly projective graphs are projective. Indeed, by theorem 10, we need to show that any idempotent homomorphism $g\colon H^{2}\to H$ is a projection. Consider a core $W$, which is incomparable with $H$, and a homomorphism $f\colon H^{2}\times W\to H$, defined by $f(x_{1},x_{2},y):=g(x_{1},x_{2})$. Since $H$ is truly projective, $f$ is a projection, and so is $g$.

Again, we state and prove the stronger version of a lower bound, parameterized by the pathwidth of an instance graph.

Theorem 19**.**

*Let $H$ be a fixed non-trivial connected core, with prime factorization $H_{1}\times\ldots\times H_{m}$. Assume there exists $i\in[m]$ such that $H_{i}$ is truly projective. Unless the SETH fails, there is no algorithm solving the Hom($H$) problem for instance graph $G$ in time $\mathcal{O}^{*}\left((|H_{i}|-\varepsilon)^{\operatorname{pw}(G)}\right)$, for any $\varepsilon>0$. *

The proof of theorem 19 is similar to the proof of 3’ (b). We start with constructing an appropriate edge gadget. We will use the following result (to avoid introducing new definitions, we state the theorem in a sightly weaker form, using the terminology used in this paper, see also [23, Theorem 8.18]).

Theorem 20** (Dörfler, [14]).**

*Let $\varphi$ be an automorphism of a connected, non-bipartite, ramified graph $H$, with the prime factorization $H_{1}\times\ldots\times H_{m}$. Then for each $i\in[m]$ there exists an automorphism $\varphi^{(i)}$ of $H_{i}$ such that $\varphi_{i}(t_{1},\dots,t_{m})\equiv\varphi^{(i)}(t_{i}).$ *

In particular, it implies the following.

Corollary 21**.**

*Let $\mu$ be an automorphism of a connected, non-trivial core $H=H_{1}\times R$, where $H_{1}$ is indecomposable and $R\not\simeq K_{1}^{*}$. Then there exist automorphisms $\mu^{(1)}:H_{1}\to H_{1}$ and $\mu^{(2)}:R\to R$ such that $\mu(t,t^{\prime})\equiv(\mu^{(1)}(t),\mu^{(2)}(t^{\prime}))$. *

Proof.

By 18, $R$ is a non-trivial core, so it admits the unique prime factorization, say $R=H_{2}\times\ldots\times H_{m}$. Therefore $H_{1}\times H_{2}\times\ldots\times H_{m}$ is the unique prime factorization of $H$. From theorem 20 we know that for each $i\in[m]$ there exists an automorphism $\varphi^{(i)}$ of $H_{i}$ such that $\mu(t_{1},\dots,t_{m})\equiv(\varphi^{(1)}(t_{1}),\dots,\varphi^{(m)}(t_{m}))$. Define $\mu^{(1)}$ by setting $\mu^{(1)}(t):=\varphi^{(1)}(t)$ for every vertex $t\in V(H_{1})$. Analogously, we define $\mu^{(2)}$ by setting $\mu^{(2)}(t_{2},\ldots,t_{m}):=(\varphi^{(2)}(t_{2}),\ldots,\varphi^{(m)}(t_{m}))$ for every vertex $(t_{2},\ldots,t_{m})$ of $R$ (for each $i\in[m]\setminus\{1\}$ we have $t_{i}\in V(H_{i})$). It is straightforward to verify that $\mu^{(1)}$ and $\mu^{(2)}$ satisfy the statement of the corollary.

In the following lemma we construct an edge gadget, that will be used in the hardness reduction. The construction is similar to the one in lemma 15, but more technically complicated.

Lemma 22**.**

Let $H=H_{1}\times R$ be a connected, non-trivial core, such that $H_{1}$ is truly projective and $R\not\simeq K_{1}^{*}$. Let $w$ be a fixed vertex of $R$. Then there exists a graph $F$ and vertices $u^{*},v^{*}$ of $F$, satisfying the following conditions:

1. (a)

for every $xy\in E(H_{1})$ there exists $f:F\to H$ such that $f(u^{*})=(x,w)$ and $f(v^{*})=(y,w)$, 2. (b)

for any $f:F\to H$ it holds that $f_{1}(u^{*})f_{1}(v^{*})\in E(H_{1})$.

Proof.

Let $E(H_{1})=\{e_{1},\ldots,e_{s}\}$ and let $e_{i}=u_{i}v_{i}$ for every $i\in[s]$ (clearly, one vertex can appear many times as some $u_{i}$ or $v_{j}$). Consider the vertices

[TABLE]

of $H_{1}^{2s}$. Let $F:=H_{1}^{2s}\times R$, and let $u^{*}:=(u,w)$ and $v^{*}:=(v,w)$. We will treat vertices $u$ and $v$ as $2s$-tuples, and vertices $u^{*}$ and $v^{*}$ as $(2s+1)$-tuples.

Observe that, if $xy\in E(H_{1})$, then, by the definition of $u^{*}$ and $v^{*}$, there exists $i\in[2s]$ such that $x=\pi_{i}(u)$ and $y=\pi_{i}(v)$. Define a function $f:V(F)\to V(H)$ as $f(x_{1},\ldots,x_{2s},r):=(\pi_{i}(x_{1},\dots,x_{2s}),r)$. Observe that this is a homomorphism, for which $f(u^{*})=f(u,w)=(x,w)$ and $f(v^{*})=f(v,w)=(y,w)$, which is exactly the condition (a) in the statement of lemma 22.

We prove (b) in two steps. First, we observe the following.

Claim**.**

*Let $\varphi:F\to H$. If for every $z\in V(H_{1})$ and $r\in V(R)$ it holds that $\varphi_{1}(z,\dots,z,r)=z$ then $\varphi_{1}(u^{*})\varphi_{1}(v^{*})\in E(H_{1})$. *

Proof of Claim. Recall that $R$ is a connected core incomparable with $H_{1}$, and $H_{1}$ is truly projective. It means that if $\varphi_{1}:H_{1}^{2s}\times R\to H_{1}$ satisfies the assumption of the claim, then it is equal to $\pi_{i}$ for some $i\in[2s]$. From the definition of $u^{*}$ and $v^{*}$ we have that $\pi_{i}(u^{*})\pi_{i}(v^{*})\in E(H_{1})$. $\blacksquare$

Note that the set $\{(z,\dots,z,r)\in F\leavevmode\nobreak\ |\leavevmode\nobreak\ z\in V(H_{1}),r\in V(R)\}$ induces in $F$ a subgraph isomorphic to $H$, let us call it $\widetilde{H}$. Let $\sigma$ be an isomorphism from $\widetilde{H}$ to $H$ defined as $\sigma(z,\dots,z,r):=(z,r)$.

Consider any homomorphism $f\colon F\to H$. We observe that $f|_{V(\widetilde{H})}$ is an isomorphism from $\widetilde{H}$ to $H$, because $H$ is a core. If $f|_{V(\widetilde{H})}\equiv\sigma$ then for every $z\in V(H)$ and $r\in V(R)$ it holds that $f_{1}(z,\dots,z,r)=\sigma_{1}(z,\dots,z,r)=z$, so, by the Claim above, we are done. If not, observe that there exists the inverse isomorphism $g:H\to\widetilde{H}$ such that $g\circ f|_{V(\widetilde{H})}$ is the identity function on $V(\widetilde{H})$. Define $\mu:=\sigma\circ g$. Observe that $\mu$ is an endomorphism of $H_{1}\times R$, so an automorphism, since $H_{1}\times R$ is a core. Also note that $(\mu\circ f):F\to H_{1}\times R$ is a homomorphism such that for every $(z,\ldots,z,r)\in V(\widetilde{H})$ it holds that

[TABLE]

so $(\mu\circ f)_{1}(z,\dots,z,z^{\prime})=z$. This means that $\mu\circ f$ satisfies the assumption of the Claim, so

[TABLE]

Clearly, for every vertex $\bar{z}$ of $F$ it holds that

[TABLE]

Note that corollary 21 implies that there exist automorphisms $\mu^{(1)}$ and $\mu^{(2)}$ of $H_{1}$ and $R$, respectively, such that for every $\bar{z}\in V(F)$ it holds that

[TABLE]

In particular, (2) and (3) imply that $\left(\mu\circ f\right)_{1}=\mu^{(1)}\circ f_{1}$. Combining this with (1) we get that

[TABLE]

Since $\mu^{(1)}$ is the automorphism of $H_{1}$, there exists the inverse automorphism $\left(\mu^{(1)}\right)^{-1}$ of $H_{1}$. Because $\left(\mu^{(1)}\right)^{-1}$ is an automorphism, (4) implies that $f_{1}(u^{*})f_{1}(v^{*})\in E(H_{1})$, which completes the proof.

Now we can proceed to the proof of theorem 19.

Proof of Theorem 19.

Since $\times$ is commutative, without loss of generality we can assume that $H_{1}$ is truly projective. Define $R:=H_{2}\times\ldots\times H_{m}$, so $H=H_{1}\times R$. Since $H_{1}$ is truly projective, it is projective, so 3’ (b) can be applied here. Hence we known that assuming the SETH, there is no algorithm which solves instances of Hom($H_{1}$) with $n$ vertices and pathwidth $t$ in time $\mathcal{O}^{*}\left((|H_{1}|-\varepsilon)^{t}\right)$, for any $\varepsilon>0$.

Let $G$ be an instance of Hom($H_{1}$) with $n$ vertices and pathwidth $t$. The construction of the instance $G^{*}$ of Hom($H$) is analogous as in the proof of 3’ (b). Let $w$ be a fixed vertex of $R$ and let $F$ be a graph obtained by calling lemma 22 for $H$ and $w$. For every vertex $z$ of $G$, we introduce to $G^{*}$ a vertex $z^{\prime}$. Then we add a copy $F_{xy}$ of $F$ for every pair of vertices $x^{\prime},y^{\prime}$, which corresponds to an edge $xy$ in $G$, and identify vertices $x^{\prime}$ and $y^{\prime}$ with vertices $u^{*}$ and $v^{*}$ of $F_{xy}$, respectively. As in the proof of 3’ (b), we observe that $G^{*}$ is a yes-instance of Hom($H$) if and only if $G$ is a yes-instance of Hom($H_{1}$). Moreover, $|G^{*}|\leqslant|F|\cdot n^{2}$ and ${\operatorname{pw}(G^{*})}\leqslant t+|F|$. Thus, if we could decide if $G^{*}\to H$ in time $\mathcal{O}^{*}\left((|H_{1}|-\varepsilon)^{{\operatorname{pw}(G^{*})}}\right)$, then we would be able to decide if $G\to H_{1}$ in time $\mathcal{O}^{*}\left((|H_{1}|-\varepsilon)^{t}\right)$. By 3’ (b), such an algorithm contradicts the SETH.

Note that combining the results from theorem 12 and theorem 19 we obtain a tight complexity bound for graphs $H$, whose largest factor is truly projective.

Corollary 23**.**

Let $H$ be a non-trivial, connected core with prime factorization $H_{1}\times\ldots\times H_{m}$ and let $H_{i}$ be the factor with the largest number of the vertices. Assume that $H_{i}$ is truly projective. Let $n$ and $t$ be, respectively, the number of vertices and the treewidth of an instance graph $G$.

1. (a)

Even if $H$ is a part of the input, the Hom($H$) problem can be solved in time $\mathcal{O}\left(|H|^{4}+|H_{i}|^{t+1}\cdot n\right)$, if a tree decomposition of $G$ of width $t$ is given. 2. (b)

Even if $H$ is fixed, there is no algorithm solving the Hom($H$) problem in time $\mathcal{O}^{*}\left((|H_{i}|-\varepsilon)^{t}\right)$ for any $\varepsilon>0$, unless the SETH fails. $\square$

5 Conclusion

Recall that in theorem 19 and corollary 23 we presented lower complexity bounds for Hom($H$) in the case that one of factors of $H$ is truly projective. In the light of corollary 3, we would like to weaken this assumption by substituting “truly projective” with “projective”. Let us discuss the possibility of obtaining such a result.

As mentioned in the introduction, a class of graphs very close to truly projective graphs was considered by Larose [32]. In the same paper, he defined and studied the so-called strongly projective graphs. A graph $H$ on at least three vertices is strongly projective, if for every connected graph $W$ on at least two vertices and every $s\geqslant 2$, the only homomorphisms $f:H^{s}\times W\to H$ satisfying $f(x,\dots,x,y)=x$ for all $x\in V(H)$ and $y\in V(W)$, are projections. Note that this definition is very similar, but more restrictive than the definition of truly projective graphs. Indeed, for truly projective graphs $H$ we restricted the homomorphisms from $H^{s}\times W$ to $H$ only for connected cores $W$, that are incomparable with $H$. Thus it is clear that every strongly projective graph is truly projective, and, as observed before, every truly projective graph is projective. Among other properties of strongly projective graphs, Larose [32, 33] shows that their recognition is decidable – note that this does not follow directly from the definition.

Let us recall some results on strongly projective graphs, as they show that many natural graphs satisfy the assumptions of theorem 19 and corollary 23. We say that graph is square-free if it does not contain a copy of $C_{4}$ as a (not necessarily induced) subgraph. Larose proved the following.

Theorem 24** (Larose [32]).**

*If $H$ is a square-free, connected, non-bipartite core, then it is strongly projective. *

Example**.**

Consider the graph $G_{B}$ on 21 vertices, shown on fig. 4 (left), it is called the Brinkmann graph [9]. It is connected, its chromatic number is 4 and its girth is 5. In particular, it is square-free. Thus by theorem 24 we know that $G_{B}$ is strongly projective. By exhaustive computer search we verified that $K_{3}\times G_{B}$ is a core. Let us consider the complexity of Hom($K_{3}\times G_{B}$) for input graphs with $n$ vertices and treewidth $t$. The straightforward dynamic programming approach from theorem 11 results in the running time $\mathcal{O}^{*}(63^{t})$. However, theorem 12 gives us a faster algorithm, whose running time is $\mathcal{O}^{*}(21^{t})$. Moreover, by corollary 23 we know that this algorithm is likely to be asymptotically optimal, i.e., there is no algorithm with running time $\mathcal{O}^{*}((21-\varepsilon)^{t})$ for any $\varepsilon>0$ and any constants $c,d$, unless the SETH fails.

A graph is said to be primitive if there is no non-trivial partition of its vertices which is invariant under all automorphisms of this graph (see e.g. [47]).

Theorem 25** (Larose [32]).**

*If $H$ is an indecomposable primitive core, then it is strongly projective. *

In particular, theorem 25 implies that Kneser graphs are strongly projective [32]. Note that Kneser graphs might have 4-cycles, so this statement does not follow from theorem 24.

Interestingly, Larose [32, 33] proved that members of all known families of projective graphs are in fact strongly projective (and thus of course truly projective). He also asked whether the same holds for all projective graphs. We recall this problem in a weaker form, which would be sufficient in our setting.

\conjstrongly

Clearly, if both corollary 3 and corollary 3 are true, there is another characterization of non-trivial indecomposable connected cores.

Observation 26**.**

*Assume that corollary 3 and corollary 3 hold. Let $H$ be a connected non-bipartite core. Then $H$ is indecomposable if and only if it is truly projective. $\square$ *

Note that section 1, corollary 23, and 26 imply the following result.

\thmeverything

We believe that the bounds from corollary 3 are tight for all connected cores. Recall that in lemma 13 and lemma 14 we showed that in order to prove tight bounds for the Hom($H$) problem, we can restrict ourselves to connected cores $H$. Observe that combining these reductions with the result of corollary 3, we obtain the following complexity bounds.

Theorem 27**.**

Assume that corollary 3 and corollary 3 hold. Let $H=H_{1}+\ldots+H_{\ell}$ be a non-trivial core and let $H_{i,1}\times\ldots\times H_{i,m_{i}}$ be the prime factorization of $H_{i}$, for every $i\in[\ell]$. Define $k:=\max_{i\in[\ell],j\in[m_{i}]}|H_{i,j}|$. Let $n$ and $t$ be, respectively, the number of vertices and the treewidth of an instance graph $G$.

1. (a)

Even if $H$ is given as an input, the Hom($H$) problem can be solved in time $\mathcal{O}(|H|^{4}+n\cdot k^{t+1}\cdot|H|)$, assuming a tree decomposition of $G$ of width $t$ is given. 2. (b)

Even if $H$ is fixed, there is no algorithm solving the Hom($H$) problem in time $\mathcal{O}^{*}\left((k-\varepsilon)^{t}\right)$ for any $\varepsilon>0$, unless the SETH fails.

Finally, let us point out one more problem, related to the ones discussed in this paper. Recall that if $H=H_{1}\times H_{2}$ is a connected, non-trivial core and $H_{1}\not\simeq K_{1}^{*},H_{2}\not\simeq K_{1}^{*}$, then $H_{1}$ and $H_{2}$ must be incomparable cores. We believe it would be interesting to know if the opposite implication holds as well. To motivate the study on this problem, we state the following conjecture.

Conjecture 1**.**

Let $H_{1}$ and $H_{2}$ be connected, indecomposable, incomparable cores. Then $H_{1}\times H_{2}$ is a core.

Note that it is straightforward to verify that if $H_{1}$ and $H_{2}$ are ramified, then their direct product $H_{1}\times H_{2}$ is ramified as well. As every core is in a particular ramified, this is a necessary condition for 1 to hold.

We confirmed the conjecture by exhaustive computer search for some small graphs. In particular, the conjecture is true for graphs $K_{3}\times H$, where $H$ is any 4-vertex-critical, triangle-free graph with at most 14 vertices [8], the Grötzsch graph (see  fig. 2), the Brinkmann graph (see fig. 4 (left)), or the Chvátal graph (see fig. 4 (right)).

Let us point out that the spirit of 1 is similar to the spirit of the recently disproved Hedetniemi’s conjecture [25, 44, 45, 48], which also asked how the properties of homomorphisms of factor graphs affect the properties of homomorphisms of their product.

Acknowledgment.

The authors are grateful to D. Marx for introducing us to the problem, and to B. Larose, C. Tardif, B. Martin, and Mi. Pilipczuk for useful comments.