Three domination number and connectivity in graphs

TL;DR

This paper introduces the concept of three domination in graphs, establishes an upper bound for the sum of this new parameter and the connectivity, and characterizes the extremal graphs achieving this bound.

Contribution

The paper proposes the novel concept of three domination in graphs and derives an upper bound for its sum with connectivity, including characterization of extremal graphs.

Findings

Established an upper bound for the sum of three domination number and connectivity.

Characterized extremal graphs where the bound is tight.

Extended domination concepts to include three domination in graph theory.

Abstract

In a graph G, a vertex dominates itself and its neighbors. A subset S of V is called a dominating set in G if every vertex in V is dominated by at least one vertex in S. The domination number gamma G is the minimum cardinality of a dominating set. A set S subset V is called a double dominating set of a graph G if every vertex in V is dominated by at least two vertices in S. The minimum cardinality of a double dominating set is called double domination number of G. In a graph G, a vertex dominates itself and its neighbors. A subset S of V is called a dominating set in G if every vertex in V is dominated by at least one vertex in S. The domination number gamma G is the minimum cardinality of a dominating set. A set S subseteq V is called a double dominating set of a graph G if every vertex in V is dominated by at least two vertices in S. The minimum cardinality of a double dominating set is called double domination number of G. The connectivity gamma G of a connected graph G is the minimum number of vertices whose removal results in a disconnected or trivial graph. In this paper, introduced the concept of three domination in graphs. and we obtain an upper bound for the sum of the three domination number and connectivity of a graph and characterize the corresponding extremal graphs.