Parameterized Complexity of Conflict-Free Graph Coloring

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.

Combinatorial Bounds for Conflict-free Coloring on Open Neighborhoods

“…2. In Section 4, we show that χ ON (G) ≤ fvs(G) + 2, where fvs(G) denotes the size of a minimum feedback vertex set of G. This bound is tight and is an improvement over the bound χ ON (G) ≤ fvs(G) + 3 by Bodlaender, Kolay and Pieterse [6]. 3.…”

“…More importantly, all our bounds are a result of constructive algorithmic procedures and hence can easily be converted into respective algorithms. We summarize our results below: 1. In Section 3, we show that χ ON (G) ≤ ⌊ 5 3 (pw(G) + 1)⌋ where pw(G) denotes the pathwidth of G. The previously best known bound in terms of pw(G) was χ ON (G) ≤ 2pw(G) + 1, implied by the results in [6].…”

“…It will be of interest to close this gap. 6. For outerplanar graphs G, the bound χ * ON (G) ≤ 4 implies a bound of χ ON (G) ≤ 5.…”

“…Since the problem is NP-hard, the parameterized aspects of the problem have been studied. The problems are fixed parameter tractable when parameterized by vertex cover number, neighborhood diversity [4], distance to cluster [5], and more recently, treewidth [6,7]. This problem has attracted special interest for graphs arising out of intersection of geometric objects, see for instance, [8,9,10].…”

Conflict-Free Coloring on Open Neighborhoods

“…Since the problem is NP-hard, the parameterized aspects of the problem have been studied. The problems are fixed parameter tractable when parameterized by vertex cover number, neighborhood diversity [4], distance to cluster, distance to threshold graphs [5], and more recently, tree-width [6,7].…”