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

2025-12-12

AI Summary

This paper investigates the Strong Conflict-Free Vertex Coloring (SCFVC) problem: given a graph (G), assign a proper (k)-vertex coloring such that, for every pair of vertices, some shortest path between them contains at most one vertex of each color—i.e., is *conflict-free*. We establish the first fixed-parameter tractability (FPT) result for SCFVC parameterized by vertex cover number ( au), designing an algorithm with runtime (2^{O( au^2)} cdot ext{poly}(n)). Furthermore, we rigorously prove that SCFVC-3—the problem of deciding whether (G) admits a strong conflict-free vertex 3-coloring—admits no polynomial kernel, even when restricted to bipartite graphs; this lower bound is shown via the cross-composition framework. Our results demonstrate that SCFVC exhibits significantly higher computational complexity than classical graph coloring and fill a key gap in the parameterized complexity theory of connectivity-constrained coloring problems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Social Networks and Social Media: Computational social science

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 $ otsubseteq$ 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.

Problem

• Decides strong conflict-free vertex-connection k-colorability for graphs
• Analyzes fixed-parameter tractability via vertex cover number parameter
• Establishes polynomial kernel lower bounds for 3-colorable bipartite graphs

Innovation

• Fixed-parameter tractable by vertex cover number
• No polynomial kernel for 3-colorable bipartite graphs
• Conflict-free shortest paths in vertex-colored graphs