Disjoint Dominating Sets with a Perfect Matching

TL;DR

This paper investigates the properties of disjoint dominating sets connected by a perfect matching, characterizes trees with specific parameters, and explores bounds in graph products and small independence number graphs.

Contribution

It characterizes trees based on the disjoint dominating sets with perfect matchings and analyzes the parameter in various graph classes.

Findings

Characterization of trees where $DD_{m}(T)$ equals certain graph parameters

Bounds for $DD_{m}(G)$ in grid graphs

Analysis of $DD_{m}(G)$ in graphs with small independence number

Abstract

In this paper, we consider dominating sets $D$ and $D'$ such that $D$ and $D'$ are disjoint and there exists a perfect matching between them. Let $DD_{\textrm{m}}(G)$ denote the cardinality of smallest such sets $D, D'$ in $G$ (provided they exist, otherwise $DD_{\textrm{m}}(G) = \infty$). This concept was introduced in [Klostermeyer et al., Theory and Application of Graphs, 2017] in the context of studying a certain graph protection problem. We characterize the trees $T$ for which $DD_{\textrm{m}}(T)$ equals a certain graph protection parameter and for which $DD_{\textrm{m}}(T) = \alpha(T)$, where $\alpha(G)$ is the independence number of $G$. We also further study this parameter in graph products, e.g., by giving bounds for grid graphs, and in graphs of small independence number.