On total domination subdivision numbers of trees

TL;DR

This paper investigates the properties of trees in graph theory related to total domination subdivision numbers, providing new characterizations for trees where this number equals 3, expanding on previous work.

Contribution

It offers new characterizations of trees with total domination subdivision number 3, building upon and extending prior characterizations.

Findings

New characterizations of trees with total domination subdivision number 3

Extended previous constructive characterizations

Enhanced understanding of total domination in trees

Abstract

A set $S$ of vertices in a graph $G$ is a total dominating set of $G$ if every vertex is adjacent to a vertex in $S$. The total domination number $\gamma_t(G)$ is the minimum cardinality of a total dominating set of $G$. The total domination subdivision number $\mbox{sd}_{\gamma_t}(G)$ of a graph $G$ is the minimum number of edges that must be subdivided (where each edge in $G$ can be subdivided at most once) in order to increase the total domination number. Haynes et al. (Discrete Math. 286 (2004) 195--202) have given a constructive characterization of trees whose total domination subdivision number is~$3$. In this paper, we give new characterizations of trees whose total domination subdivision number is 3.