Complexity of planar signed graph homomorphisms to cycles

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14330%2F20%3A00118538" target="_blank" >RIV/00216224:14330/20:00118538 - isvavai.cz</a>

Result on the web

<a href="https://doi.org/10.1016/j.dam.2020.03.029" target="_blank" >https://doi.org/10.1016/j.dam.2020.03.029</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.dam.2020.03.029" target="_blank" >10.1016/j.dam.2020.03.029</a>

Result language

angličtina

Original language name

Complexity of planar signed graph homomorphisms to cycles

Original language description

We study homomorphism problems of signed graphs from a computational point of view. A signed graph is an undirected graph where each edge is given a sign, positive or negative. An important concept when studying signed graphs is the operation of switching at a vertex, which is to change the sign of each incident edge. A homomorphism of a graph is a vertex-mapping that preserves the adjacencies; in the case of signed graphs, we also preserve the edge-signs. Special homomorphisms of signed graphs, called s-homomorphisms, have been studied. In an s-homomorphism, we allow, before the mapping, to perform any number of switchings on the source signed graph. The concept of s-homomorphisms has been extensively studied, and a full complexity classification (polynomial or NP-complete) for s-homomorphism to a fixed target signed graph has recently been obtained. Nevertheless, such a dichotomy is not known when we restrict the input graph to be planar, not even for non-signed graph homomorphisms. We show that deciding whether a (non-signed) planar graph admits a homomorphism to the square C-t(2) of a cycle with t &gt;= 6, or to the circular clique K-4t(/()2t(-1)) with t &gt;= 2, are NP-complete problems. We use these results to show that deciding whether a planar signed graph admits an s-homomorphism to an unbalanced even cycle is NP-complete. (A cycle is unbalanced if it has an odd number of negative edges). We deduce a complete complexity dichotomy for the planar s-homomorphism problem with any signed cycle as a target. We also study further restrictions involving the maximum degree and the girth of the input signed graph. We prove that planar s-homomorphism problems to signed cycles remain NP-complete even for inputs of maximum degree 3 (except for the case of unbalanced 4-cycles, for which we show this for maximum degree 4). We also show that for a given integer g, the problem for signed bipartite planar inputs of girth g is either trivial or NP-complete. (C) 2020 Published by Elsevier B.V.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Project

—

Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2020

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Discrete Applied Mathematics

ISSN

0166-218X

e-ISSN

1872-6771

Volume of the periodical

284

Issue of the periodical within the volume

30 September 2020

Country of publishing house

US - UNITED STATES

Number of pages

13

Pages from-to

166-178

UT code for WoS article

000543418800016

EID of the result in the Scopus database

2-s2.0-85082831519