Parameterized Algorithms for Conflict-free Colorings of Graphs

TL;DR

This paper investigates the conflict-free coloring of graphs based on neighborhoods, establishing fixed-parameter tractability results, approximation algorithms, and polynomial-time solutions for specific graph classes, advancing understanding of the problem's complexity.

Contribution

It extends fixed-parameter tractability results for conflict-free coloring to new parameters and graph classes, and provides polynomial algorithms for certain graph types.

Findings

Both conflict-free coloring problems are FPT when parameterized by cluster vertex deletion number.

Polynomial-time algorithms exist for cographs and split graphs for specific cases.

Interval graphs can be conflict-free colored with at most four colors.

Abstract

In this paper, we study the conflict-free coloring of graphs induced by neighborhoods. A coloring of a graph is conflict-free if every vertex has a uniquely colored vertex in its neighborhood. The conflict-free coloring problem is to color the vertices of a graph using the minimum number of colors such that the coloring is conflict-free. We consider both closed neighborhoods, where the neighborhood of a vertex includes itself, and open neighborhoods, where a vertex does not included in its neighborhood. We study the parameterized complexity of conflict-free closed neighborhood coloring and conflict-free open neighborhood coloring problems. We show that both problems are fixed-parameter tractable (FPT) when parameterized by the cluster vertex deletion number of the input graph. This generalizes the result of Gargano et al.(2015) that conflict-free coloring is fixed-parameter tractable parameterized by the vertex cover number. Also, we show that both problems admit an additive constant approximation algorithm when parameterized by the distance to threshold graphs. We also study the complexity of the problem on special graph classes. We show that both problems can be solved in polynomial time on cographs. For split graphs, we give a polynomial time algorithm for closed neighborhood conflict-free coloring problem, whereas we show that open neighborhood conflict-free coloring is NP-complete. We show that interval graphs can be conflict-free colored using at most four colors.