List Locally Surjective Homomorphisms in Hereditary Graph Classes
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866916086271705088 |
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| author | Dvořák, Pavel Krawczyk, Monika Masařík, Tomáš Novotná, Jana Rzążewski, Paweł Żuk, Aneta |
| author_facet | Dvořák, Pavel Krawczyk, Monika Masařík, Tomáš Novotná, Jana Rzążewski, Paweł Żuk, Aneta |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_12438 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | List Locally Surjective Homomorphisms in Hereditary Graph Classes Dvořák, Pavel Krawczyk, Monika Masařík, Tomáš Novotná, Jana Rzążewski, Paweł Żuk, Aneta Data Structures and Algorithms Computational Complexity 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). |
| title | List Locally Surjective Homomorphisms in Hereditary Graph Classes |
| topic | Data Structures and Algorithms Computational Complexity |
| url | https://arxiv.org/abs/2202.12438 |