Conflict-Free Coloring: Graphs of Bounded Clique Width and Intersection Graphs

TL;DR

The CFON* problem is fixed-parameter tractable with respect to the combined parameters clique width and the solution size and the problem is NP-complete even on planar graphs.

Abstract

A conflict-free coloring of a graph $G$ is a (partial) coloring of its vertices such that every vertex $u$ has a neighbor whose assigned color is unique in the neighborhood of $u$. There are two variants of this coloring, one defined using the open neighborhood and one using the closed neighborhood. For both variants, we study the problem of deciding whether the conflict-free coloring of a given graph $G$ is at most a given number $k$. In this work, we investigate the relation of clique-width and minimum number of colors needed (for both variants) and show that these parameters do not bound one another. Moreover, we consider specific graph classes, particularly graphs of bounded clique-width and types of intersection graphs, such as distance hereditary graphs, interval graphs and unit square and disk graphs. We also consider Kneser graphs and split graphs. We give (often tight) upper and lower bounds and determine the complexity of the decision problem on these graph classes, which improve some of the results from the literature. Particularly, we settle the number of colors needed for an interval graph to be conflict-free colored under the open neighborhood model, which was posed as an open problem.

Figures (5)

• Figure 1: $G_3$ (left) and $G_4$ (right) have clique-width $3$ but cannot be CFCN colored with $2$ and $3$ colors, respectively. Each $G_i$,$i\geq 2$ stands for a copy of the graph $G_i$. Every vertex in an ellipse is adjacent to every vertex that is connected to that ellipse.
• Figure 2: Bull graph $G$ with $\chi^*_{CN}(G)=2$.
• Figure 3: A bipartite distance hereditary graph $G_4$ with $\chi_{ON}(G_4) \geq 4$.
• Figure 4: A block graph $G$ with $\chi^*_{ON}(G) = 3$.
• Figure 5: On the left hand side, we have the graph $G'$, and on the right hand side we have an interval representation of $G$, a graph in which $\chi_{ON}(G){} > 3$. The graph $G$ is obtained by adding two true twins each to the vertices $u,v,w,u^\star,v^\star$ of $G'$ and adding three true twins each to the vertices $u',u",v',v",w',w"$ of $G'$.

Theorems & Definitions (31)

• definition thmcounterdefinition: Conflict-Free Coloring
• definition thmcounterdefinition: Conflict-Free Coloring -- Full Coloring Variant
• remark thmcounterremark
• definition thmcounterdefinition: Clique-width courcellecw
• theorem 1.1
• theorem 1.2
• proof
• theorem 1.3
• definition thmcounterdefinition: Cograph corneil1981complement
• definition thmcounterdefinition: Distance hereditary graph DistHere_Howorka