Parameterized Complexity of Conflict-Free Graph Coloring

Given a graph $G$, a $q$-open neighborhood conflict-free coloring or $q$-ONCF-coloring is a vertex coloring $c\colon V(G) \rightarrow \{1,2,\ldots,q\}$ such that for each vertex $v \in V(G)$ 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 fixed-parameter tractable algorithms for these problems when parameterized by treewidth. We improve the existing upper bounds, and also provide lower bounds on the running time under the exponential time hypothesis and the strong exponential time hypothesis. Second, we study the kernelization complexity of both problems, using vertex cover as the parameter. We show that both $(q \geq 2)$-ONCF-coloring and $(q \geq 3)$-CNCF-coloring cannot have polynomial kernels when parameterized by the size of a vertex cover unless ${\mathsf{NP \subseteq\- coNP/poly}}$. 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 $\chi_{ON}(G)$ and $\chi_{CN}(G)$ to be the minimum number of colors required to ONCF-color and CNCF-color $G$, respectively. Upper bounds on $\chi_{CN}(G)$ 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 $\chi_{ON}(G)$ with respect to minimum vertex cover size was known. We provide tight bounds for $\chi_{ON}(G)$ with respect to minimum vertex cover size. Also, we provide the first upper bounds on $\chi_{ON}(G)$ with respect to minimum feedback vertex set size and treewidth.