Graphs with Large Disjunctive Total Domination Number

TL;DR

This paper introduces and studies the disjunctive total domination number, a relaxation of the total domination number, providing bounds for connected graphs with minimum degree at least 2 and characterizing extremal cases.

Contribution

It defines the disjunctive total domination number and establishes new upper bounds for it in graphs with minimum degree at least 2, extending known results for total domination.

Findings

Disjunctive total domination number is at most half the number of vertices for certain graphs.

Characterization of extremal graphs where the bound is tight.

Extension of known bounds from total domination to disjunctive total domination.

Abstract

Let $G$ be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, $\gamma_t(G)$. A set $S$ of vertices in $G$ is a disjunctive total dominating set of $G$ if every vertex is adjacent to a vertex of $S$ or has at least two vertices in $S$ at distance $2$ from it. The disjunctive total domination number, $\gamma^d_t(G)$, is the minimum cardinality of such a set. We observe that $\gamma^d_t(G) \le \gamma_t(G)$. Let $G$ be a connected graph on $n$ vertices with minimum degree $\delta$. It is known [J. Graph Theory 35 (2000), 21--45] that if $\delta \ge 2$ and $n \ge 11$, then $\gamma_t(G) \le 4n/7$. Further [J. Graph Theory 46 (2004), 207--210] if $\delta \ge 3$, then $\gamma_t(G) \le n/2$. We prove that if $\delta \ge 2$ and $n \ge 8$, then $\gamma^d_t(G) \le n/2$ and we characterize the extremal graphs.