Parameterized complexity of conflict-free graph coloring

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Abstract

Given a graph G, a q-open neighborhood conflict-free coloring or q-ONCF-coloring is a vertex coloring c: V(G) → { 1, 2, …, q} such that for each vertex v∈ V(G) there is a vertex in N(v) that is uniquely colored from the rest of the vertices in N(v). When we replace N(v) by the closed neighborhood N[v], then we call such a coloring a q-closed neighborhood conflict-free coloring or simply q-CNCF-coloring. In this paper, we study the NP-hard decision questions of whether for a constant q an input graph has a q-ONCF-coloring or a q-CNCF-coloring. We will study these two problems in the parameterized setting. First of all, we study running time bounds on FPT-algorithms for these problems, when parameterized by treewidth. We improve the existing upper bounds, and also provide lower bounds on the running time under ETH and SETH. Secondly, we study the kernelization complexity of both problems, using vertex cover as the parameter. We show that both (q≥ 2) -ONCF-coloring and (q≥ 3) -CNCF-coloring cannot have polynomial kernels when parameterized by the size of a vertex cover unless NP⊆ coNP/poly. On the other hand, we obtain a polynomial kernel for 2-CNCF-coloring parameterized by vertex cover. We conclude the study with some combinatorial results. Denote χON(G) and χCN(G) to be the minimum number of colors required to ONCF-color and CNCF-color G, respectively. Upper bounds on χCN(G) with respect to structural parameters like minimum vertex cover size, minimum feedback vertex set size and treewidth are known. To the best of our knowledge only an upper bound on χON(G) with respect to minimum vertex cover size was known. We provide tight bounds for χON(G) with respect to minimum vertex cover size. Also, we provide the first upper bounds on χON(G) with respect to minimum feedback vertex set size and treewidth.

Original languageEnglish
Title of host publicationAlgorithms and Data Structures - 16th International Symposium, WADS 2019, Proceedings
EditorsZachary Friggstad, Mohammad R. Salavatipour, Jörg-Rüdiger Sack
Place of PublicationCham
PublisherSpringer
Pages168-180
Number of pages13
ISBN (Electronic)978-3-030-24766-9
ISBN (Print)978-3-030-24765-2
DOIs
Publication statusPublished - 12 Jul 2019
Event16th International Symposium on Algorithms and Data Structures, WADS 2019 - Edmonton, Canada
Duration: 5 Aug 20197 Aug 2019

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume11646 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference16th International Symposium on Algorithms and Data Structures, WADS 2019
CountryCanada
CityEdmonton
Period5/08/197/08/19

Keywords

  • Combinatorial bounds
  • Conflict-free coloring
  • Fixed-parameter tractability
  • Kernelization

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