Characterizations of the graphs with dominating parameters

TL;DR

This paper explores various domination parameters in graphs, providing characterizations and solutions to open problems related to domination, total domination, and isolate sets, thereby advancing understanding of graph domination theory.

Contribution

It offers new characterizations of graphs with specific domination parameters and resolves open problems from prior research.

Findings

Characterizations of graphs with given domination parameters

Solutions to open problems on domination and isolate sets

Enhanced understanding of domination parameters in graph theory

Abstract

A subset $S$ of vertices of $G$ is a \textit{dominating set} of $G$ if every vertex in $V(G)-S$ has a neighbor in $S$. The \textit{domination number} \(\gamma(G)\) is the minimum cardinality of a dominating set of $G$. A dominating set $S$ is a \textit{total dominating set} if $N(S)$=$V$ where $N(S)$ is the neighbor of $S$. The \textit{total domination number} \(\gamma_t(G)\) equals the minimum cardinality of a total dominating set of $G$. A set $D$ is an \textit{isolate set} if the induced subgragh $G[D]$ has at least one isolated vertex. The \textit{isolate number} \(i_0(G)\) is the minimum cardinality of a maximal isolate set. In this paper we study these parameters and answer open problems proposed by Hamid et al. in 2016.