Common domination perfect graphs

TL;DR

This paper characterizes common domination perfect graphs by identifying ten forbidden induced subgraphs, expanding the understanding of domination and independence properties in graph theory.

Contribution

It introduces the concept of common domination perfect graphs and provides a forbidden subgraph characterization for this class.

Findings

Characterization of common domination perfect graphs

Identification of ten forbidden induced subgraphs

Extension of domination and independence concepts

Abstract

A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex that does not belong to $S$ is adjacent to a vertex in $S$. The domination number $\gamma(G)$ of $G$ is the minimum cardinality of a dominating set of $G$. The common independence number $\alpha_c(G)$ of $G$ is the greatest integer $r$ such that every vertex of $G$ belongs to some independent set of cardinality at least~$r$. The common independence number is squeezed between the independent domination number $i(G)$ and the independence number $\alpha(G)$ of $G$, that is, $\gamma(G) \le i(G) \le \alpha_c(G) \le \alpha(G)$. A graph $G$ is domination perfect if $\gamma(H) = i(H)$ for every induced subgraph $H$ of $G$. We define a graph $G$ as common domination perfect if $\gamma(H) = \alpha_c(H)$ for every induced subgraph $H$ of $G$. We provide a characterization of common domination perfect graphs in terms of ten forbidden induced subgraphs.