Paper
The parameterized complexity of Strong Conflict-Free Vertex-Connection Colorability
Authors
Carl Feghali, Hoang-Oanh Le, Van Bang Le
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 -colorable if it admits a (proper) vertex -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 and an integer , whether is strongly conflict-free -colorable is fixed-parameter tractable when parameterized by the vertex cover number. But under the standard complexity-theoretic assumption NP coNP/poly, deciding, for a given graph , whether is strongly conflict-free -colorable does not admit a polynomial kernel, even for bipartite graphs. This kernel lower bound is in stark contrast to the ordinal -Coloring problem which is known to admit a polynomial kernel when parameterized by the vertex cover number.