On Perfect and Quasiperfect Domination in Graphs

TL;DR

This paper investigates the properties and existence of graphs with short quasiperfect domination chains, focusing on cases where the chain length is minimal, including classes like cographs and claw-free graphs.

Contribution

It characterizes graphs with minimal quasiperfect domination chains and explores their properties and realizations, extending the understanding of domination concepts in graph theory.

Findings

Graphs with short quasiperfect domination chains include cographs and claw-free graphs.

The paper establishes conditions for the existence of graphs where 2(G) = (G).

It identifies extremal graphs with specific domination properties.

Abstract

A subset $S\subseteq V$ in a graph $G=(V,E)$ is a $k$-quasiperfect dominating set (for $k\geq 1$) if every vertex not in $S$ is adjacent to at least one and at most $k$ vertices in $S$. The cardinality of a minimum $k$-quasiperfect dominating set in $G$ is denoted by $\gamma_ {\stackrel{}{1k}}(G)$. Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept and allow us to construct a decreasing chain of quasiperfect dominating numbers $ n \ge \gamma_ {\stackrel{}{11}}(G) \ge \gamma_ {\stackrel{}{12}}(G)\ge \ldots \ge \gamma_ {\stackrel{}{1\Delta}}(G)=\gamma(G)$ in order to indicate how far is $G$ from being perfectly dominated. In this paper we study properties, existence and realization of graphs for which the chain is short, that is, $\gamma_ {\stackrel{}{12}}(G)=\gamma (G)$. Among them, one can find cographs, claw-free graphs and graphs with extremal values of $\Delta(G)$.