Equality in a Linear Vizing-Like Relation that Relates the Size and Total Domination Number of a Graph

TL;DR

This paper characterizes the extremal graphs that achieve equality in a linear relation connecting the size and total domination number, extending known bounds in graph theory.

Contribution

It provides a complete characterization of graphs where the size equals the maximum degree times the difference between order and total domination number.

Findings

Identifies extremal graphs satisfying the equality

Extends previous bounds on graph size and domination number

Provides structural insights into such extremal graphs

Abstract

Let $G$ be a graph each component of which has order at least 3, and let $G$ have order $n$, size $m$, total domination number $\gamma_t$ and maximum degree $\Delta(G)$. Let $\Delta = 3$ if $\Delta(G) = 2$ and $\Delta = \Delta (G)$ if $\Delta(G) \ge 3$. It is known [J. Graph Theory 49 (2005), 285--290; J. Graph Theory 54 (2007), 350--353] that $m \le \Delta (n- \gamma_t)$. In this paper we characterize the extremal graphs $G$ satisfying $m = \Delta (n- \gamma_t)$.