Domination on Vertex-weighted Graphs Induce by a Coloring
Mar\'ia A. Garrido-Vizuete, Mucuy-kak Guevara, Alberto M\'arquez,, Rafael Robles

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…
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph Labeling and Dimension Problems · Scheduling and Optimization Algorithms
