The parameterized complexity of Strong Conflict-Free Vertex-Connection Colorability

TL;DR

This paper investigates the computational complexity of a new graph coloring variant called strong conflict-free vertex-connection, establishing fixed-parameter tractability results and kernelization lower bounds for various parameters and graph classes.

Contribution

It introduces the parameterized complexity analysis of strong conflict-free vertex-connection coloring, including FPT algorithms and kernelization lower bounds.

Findings

Deciding strong conflict-free $k$-colorability is fixed-parameter tractable by vertex cover number.

No polynomial kernel exists for testing 3-colorability under standard complexity assumptions.

Contrast established with polynomial kernel results for ordinal $k$-Coloring.

Abstract

This paper continues the study of a new variant of graph coloring with a connectivity constraint recently introduced by Hsieh et al. [COCOON 2024]. A path in a vertex-colored graph is called conflict-free if there is a color that appears exactly once on its vertices. A connected graph is said to be strongly conflict-free vertex-connection $k$-colorable if it admits a (proper) vertex $k$-coloring such that any two distinct vertices are connected by a conflict-free shortest path. Among others, we show that deciding, for a given graph $G$ and an integer $k$, whether $G$ is strongly conflict-free $k$-colorable is fixed-parameter tractable when parameterized by the vertex cover number. But under the standard complexity-theoretic assumption NP $\not\subseteq$ coNP/poly, deciding, for a given graph $G$, whether $G$ is strongly conflict-free $3$-colorable does not admit a polynomial kernel, even for bipartite graphs. This kernel lower bound is in stark contrast to the ordinal $k$-Coloring problem which is known to admit a polynomial kernel when parameterized by the vertex cover number.