List Locally Surjective Homomorphisms in Hereditary Graph Classes

TL;DR

This work studies the complexity of the list locally surjective graph homomorphism problem LLSHom($H$) within hereditary ($F$-free) graph classes. It establishes a dichotomy: if $H \\in \\mathcal{H}_{\\mathrm{poly}}$, the problem is polynomial-time solvable; otherwise, subexponential solvability in $F$-free graphs depends on forbidding graphs in the family $\\mathcal{S}$ (forests whose components are paths or subdivided claws). For $H \\in \\{P_3, C_4\\}$, a subexponential algorithm exists for every $F \\in \\mathcal{S}$, achieving time $2^{\\mathcal{O}((n \\log n)^{2/3})}$, while for other $H$ there is ETH-based hardness unless $F$ is restricted. The paper also provides a general reduction framework using the associated bipartite graph $H^* = H \\times K_2$ to transfer hardness across bipartite and non-bipartite targets, and introduces gadget-based lifts to extend hardness from $P_3$ to general $H$ containing a $P_3$. Overall, the results significantly advance understanding of how forbidding induced subgraphs and the target graph structure together constrain the complexity of list locally surjective homomorphisms, with implications for subexponential algorithms in restricted graph classes.

Abstract

A locally surjective homomorphism from a graph $G$ to a graph $H$ is an edge-preserving mapping from $V(G)$ to $V(H)$ that is surjective in the neighborhood of each vertex in $G$. In the list locally surjective homomorphism problem, denoted by LLSHom($H$), the graph $H$ is fixed and the instance consists of a graph $G$ whose every vertex is equipped with a subset of $V(H)$, called list. We ask for the existence of a locally surjective homomorphism from $G$ to $H$, where every vertex of $G$ is mapped to a vertex from its list. In this paper, we study the complexity of the LLSHom($H$) problem in $F$-free graphs, i.e., graphs that exclude a fixed graph $F$ as an induced subgraph. We aim to understand for which pairs $(H,F)$ the problem can be solved in subexponential time. We show that for all graphs $H$, for which the problem is NP-hard in general graphs, it cannot be solved in subexponential time in $F$-free graphs unless $F$ is a bounded-degree forest or the ETH fails. The initial study reveals that a natural subfamily of bounded-degree forests $F$ that might lead to some tractability results is the family $\mathcal S$ consisting of forests whose every component has at most three leaves. In this case, we exhibit the following dichotomy theorem: besides the cases that are polynomial-time solvable in general graphs, the graphs $H \in \{P_3,C_4\}$ are the only connected ones that allow for a subexponential-time algorithm in $F$-free graphs for every $F \in \mathcal S$ (unless the ETH fails).

Figures (8)

• Figure 1: An example of variable gadgets connected to a clause gadget for the case $H=K_{1,3}$. The variable gadget is depicted by blue color, clause gadget by black color, green are connections between each variable and clause gadgets, and red is the connection representing an occurrence of a variable in the clause (in this example, a positive occurrence).
• Figure 2: An example of variable gadgets connected to a clause gadget for the case ${H=P_4}$. The variable gadget is depicted by blue color, clause gadget by black color, green are connections between each variable and clause gadgets, and red is the connection representing an occurrence of a variable in the clause (in this example, a positive occurrence).
• Figure 3: An example of variable gadgets connected to a clause gadget for the case ${H=K_2^{\circ\circ}}$. The variable gadget is depicted by blue color, clause gadget by black color, green are connections between each variable and clause gadgets, and red is the connection representing an occurrence of a variable in the clause (in this example, a positive occurrence).
• Figure 4: An example of constuction of $(G,L)$ shown on one edge $w,w'$ of $(G',L')$. The added gadgets are attached using red dash-dotted edges.
• Figure 5: An example of variable gadgets connected to a clause gadget $\textit{Cls}(c)$ for $p = 1$ and $c = \neg x_1 \vee x_2 \vee x_3$ (thus, $\textit{Cls}(c)$ has two positive and one negative arm). Since the graph $G'$ is bipartite, vertices of one bipartition class (with the list $\{1,3\}$) are depicted as boxes and vertices from the other one as disks. The bold numbers in the lists $\{1,3\}$ represent images of the vertices under a homomorphism $h: (G,L) \xrightarrow{s} P_3$ created from a truth assignment of $\Phi$ such that $x_1$ and $x_3$ are true and $x_2$ is false.

Theorems & Definitions (39)

• Theorem 1.1
• Theorem 1.2
• Corollary 2.1
• proof
• Definition 2.2
• Lemma 2.3
• proof
• Corollary 2.4
• proof
• Corollary 2.5