Disjunctive Total Domination in Graphs

TL;DR

This paper introduces and studies the disjunctive total domination number, a relaxation of the total domination number, providing bounds for connected graphs and characterizing extremal cases, especially in claw-free graphs.

Contribution

It defines the disjunctive total domination number, establishes upper bounds for connected graphs, and improves bounds for claw-free graphs, including characterizations of extremal graphs.

Findings

For connected graphs of order n ≥ 8, γ^d_t(G) ≤ 2(n-1)/3.

For connected claw-free graphs of order n > 10, γ^d_t(G) ≤ 4n/7.

Characterization of extremal graphs achieving bounds.

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 distance2 from it. The disjunctive total domination number, $\gamma^d_t(G)$, is the minimum cardinality of such a set. We observe that $\gamma^d_t(G) \le \gamma_t(G)$. We prove that if $G$ is a connected graph of order$n \ge 8$, then $\gamma^d_t(G) \le 2(n-1)/3$ and we characterize the extremal graphs. It is known that if $G$ is a connected claw-free graph of order$n$, then $\gamma_t(G) \le 2n/3$ and this upper bound is tight for arbitrarily large$n$. We show this upper bound can be improved significantly for the disjunctive total domination number. We show that if $G$ is a connected claw-free graph of order$n > 10$, then $\gamma^d_t(G) \le 4n/7$ and we characterize the graphs achieving equality in this bound.