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Domination on Vertex-weighted Graphs Induce by a Coloring

María A. Garrido-Vizuete, Mucuy-kak Guevara, Alberto Márquez, Rafael Robles

TL;DR

This work defines up-color domination on vertex-colored graphs, where a dominating vertex must have a strictly larger color than the dominated one and color-0 vertices cannot dominate. It establishes NP-completeness for computing the up-color domination number $_{uc}(G,c)$ even on bipartite graphs with three colors, while giving a linear-time algorithm for trees, and introduces the chromatic up-color domination number $_{uc}(G)$ with NP-hardness results via 3-SAT reductions. The paper also studies the weight version, proving NP-completeness for both the weighted dominating set problem and the overall weight $(G)$, and provides a polynomial-time $O(n^2 )$-time algorithm for trees. Throughout, it relates these notions to classical domination, coloring, Grundy, and Roman domination, and derives tight bounds for several graph families. The results highlight rich interactions between coloring structure and domination, with clear implications for algorithm design on trees and for computational hardness on general graphs.

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.

Domination on Vertex-weighted Graphs Induce by a Coloring

TL;DR

This work defines up-color domination on vertex-colored graphs, where a dominating vertex must have a strictly larger color than the dominated one and color-0 vertices cannot dominate. It establishes NP-completeness for computing the up-color domination number even on bipartite graphs with three colors, while giving a linear-time algorithm for trees, and introduces the chromatic up-color domination number with NP-hardness results via 3-SAT reductions. The paper also studies the weight version, proving NP-completeness for both the weighted dominating set problem and the overall weight , and provides a polynomial-time -time algorithm for trees. Throughout, it relates these notions to classical domination, coloring, Grundy, and Roman domination, and derives tight bounds for several graph families. The results highlight rich interactions between coloring structure and domination, with clear implications for algorithm design on trees and for computational hardness on general graphs.

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.

Paper Structure

This paper contains 5 sections, 18 theorems, 12 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Definitions and preliminaries
  3. Up--color domination number
  4. Up--color domination weight
  5. Conclusions and open problems

Key Result

Lemma 2.1

Given a pair $(G,c)$, $\gamma_{uc}(G,c) \geq \max \{ |c^{-1}(i)|, \gamma(G) \}$, where $i$ represents the largest color with non--empty inverse image.

Figures (14)

  • Figure 1: Theorem \ref{['th:npcd']}.
  • Figure 2: The gadget for the variables. Theorem \ref{['th:npchi']}.
  • Figure 3: The gadget for the clauses. Theorem \ref{['th:npchi']}.
  • Figure 4: An example of integrated gadgets. Theorem \ref{['th:npchi']}.
  • Figure 5: Three optimal colorings for the same graph.
  • ...and 9 more figures

Theorems & Definitions (34)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 3.1
  • proof
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • Remark 3.1
  • Theorem 3.4
  • proof
  • ...and 24 more