| Title: |
Complexity of planar signed graph homomorphisms to cycles |
| Authors: |
Dross, François; Foucaud, Florent; Mitsou, Valia; Ochem, Pascal; Pierron, Théo |
| Contributors: |
Uniwersytet Warszawski Polska = University of Warsaw Poland = Université de Varsovie Pologne (UW); Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis (I3S); Université Nice Sophia Antipolis (1965 - 2019) (UNS)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UniCA); Laboratoire Bordelais de Recherche en Informatique (LaBRI); Université de Bordeaux (UB)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)-Centre National de la Recherche Scientifique (CNRS); Laboratoire d'Informatique Fondamentale d'Orléans (LIFO); Université d'Orléans (UO)-Institut National des Sciences Appliquées - Centre Val de Loire (INSA CVL); Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA); Institut de Recherche en Informatique Fondamentale (IRIF (UMR_8243)); Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité); Algorithmes, Graphes et Combinatoire (ALGCO); Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier (LIRMM); Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS); Masaryk University Brno = Masarykova univerzita Brno = Université Masaryk Brno (MU / MUNI); ANR-17-CE40-0022,HOSIGRA,Homomorphismes de graphes signés(2017) |
| Source: |
ISSN: 0166-218X ; Discrete Applied Mathematics ; https://hal.science/hal-02990576 ; Discrete Applied Mathematics, 2020, 284, pp.166-178. ⟨10.1016/j.dam.2020.03.029⟩. |
| Publisher Information: |
CCSD; Elsevier |
| Publication Year: |
2020 |
| Subject Terms: |
planar graph; graph homomorphism; edge-coloured graph; signed graph; [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] |
| Description: |
International audience ; 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 2 t of a cycle with t 6, or to the circular clique K 4t/(2t−1) with t 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. |
| Document Type: |
article in journal/newspaper |
| Language: |
English |
| Relation: |
info:eu-repo/semantics/altIdentifier/arxiv/1907.03266; ARXIV: 1907.03266 |
| DOI: |
10.1016/j.dam.2020.03.029 |
| Availability: |
https://hal.science/hal-02990576; https://hal.science/hal-02990576v1/document; https://hal.science/hal-02990576v1/file/1907.03266.pdf; https://doi.org/10.1016/j.dam.2020.03.029 |
| Rights: |
https://about.hal.science/hal-authorisation-v1/ ; info:eu-repo/semantics/OpenAccess |
| Accession Number: |
edsbas.9C51996 |
| Database: |
BASE |