Domination on Vertex-weighted Graphs Induce by a Coloring

TL;DR

This paper introduces a new vertex domination concept in colored graphs based on vertex weights, proves its computational hardness, and provides efficient algorithms for trees, connecting it to classical domination and coloring.

Contribution

It defines up-color domination, analyzes its NP-completeness, and develops polynomial algorithms for trees, advancing understanding of domination in weighted, colored graphs.

Findings

NP-complete for bipartite graphs with three colors

Polynomial algorithms for trees

Established links to classical domination and coloring

Abstract

This paper introduces the concept of domination in the context of colored graphs (where each color assigns a weight to the vertices of its class), termed up-color domination, where a vertex dominating another must be heavier than the other. That idea defines, on one hand, a new parameter measuring the size of minimal dominating sets satisfying specific constraints related to vertex colors. The paper proves that the optimization problem associated with that concept is an NP-complete problem, even for bipartite graphs with three colors. On the other hand, a weight-based variant, the up-color domination weight, is proposed, further establishing its computational hardness. The work also explores the relationship between up-color domination and classical domination and coloring concepts. Efficient algorithms for trees are developed that use their acyclic structure to achieve polynomial-time solutions.