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Unique paired vs edge-vertex minimum dominating sets in trees

Mateusz Miotk, Michał Zakrzewski, Paweł Żyliński

TL;DR

Problem: characterize trees where the minimum edge-vertex dominating set (ev-domination) is unique and relate it to trees with a unique minimum paired dominating set. Approach: derive a structural lemma on ev-sets, introduce a 'twinning' operation to transform one ev-set into another, and analyze implications for trees. Contributions: prove the equivalence between the classes of trees with unique $\gamma_{ev}$-sets and unique $\gamma_{pr}$-sets; show that in such trees all minimum ev-sets are supported on the same vertex subset, and establish related corollaries linking the uniqueness of one domination type to the other. Significance: provides a parallel, constructive framework to classify trees with unique domination notions and clarifies the relationship between edge-vertex and paired domination in trees.

Abstract

We prove that the class of trees with unique minimum edge-vertex dominating sets is equivalent to the class of trees with unique minimum paired dominating sets.

Unique paired vs edge-vertex minimum dominating sets in trees

TL;DR

Problem: characterize trees where the minimum edge-vertex dominating set (ev-domination) is unique and relate it to trees with a unique minimum paired dominating set. Approach: derive a structural lemma on ev-sets, introduce a 'twinning' operation to transform one ev-set into another, and analyze implications for trees. Contributions: prove the equivalence between the classes of trees with unique -sets and unique -sets; show that in such trees all minimum ev-sets are supported on the same vertex subset, and establish related corollaries linking the uniqueness of one domination type to the other. Significance: provides a parallel, constructive framework to classify trees with unique domination notions and clarifies the relationship between edge-vertex and paired domination in trees.

Abstract

We prove that the class of trees with unique minimum edge-vertex dominating sets is equivalent to the class of trees with unique minimum paired dominating sets.

Paper Structure

This paper contains 2 sections, 6 theorems, 1 figure.

Table of Contents

  1. Introduction
  2. Main Results

Key Result

Theorem 1.1

HHH11 If $G$ is a graph with no isolated vertex, then $2\gamma_{\rm ev}({G})=\gamma_{\rm pr}({G})$.

Figures (1)

  • Figure 1: The graph $G$ has a unique $\gamma_{\rm pr}$-set (gray vertices), whereas its $\gamma_{\rm ev}$-set is not unique: there are two such sets, but both spanned on the same set of (gray) vertices.

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Corollary 2.3
  • Corollary 2.4