Characterization of Well-Totally Dominated Trees

TL;DR

This paper characterizes well-totally dominated trees, providing descriptive and constructive characterizations, and shows that verifying the WTD property in trees can be done efficiently in polynomial time.

Contribution

It offers the first comprehensive characterizations of well-totally dominated trees and demonstrates a polynomial-time verification method.

Findings

Two characterizations of WTD trees: descriptive and constructive

Verification of WTD property in trees is polynomial-time feasible

Provides insights into the structure of well-totally dominated trees

Abstract

Let $G$ be a graph with no isolated vertices. A set of vertices $S$ is a total dominating set (TDS) if every vertex in $G$ is adjacent to at least one vertex in $S$. We say $G$ is well-totally dominated (WTD) if every minimal TDS has the same size. In this paper, we present two characterizations of well-totally dominated trees, one being descriptive and the other being constructive. In particular, our characterizations imply that it takes only polynomial time to verify whether a given tree is WTD.