Characterization of Well-Totally Dominated Trees

TL;DR

The paper investigates well-totally dominated (WTD) trees, where every minimal total dominating set (TDS) has the same size, addressing the computational challenge posed by the NP-hardness of generic total domination problems. It offers two complementary characterizations: a descriptive approach using a red/blue interior-forest decomposition, and a constructive approach via a whisker-adding operator that builds all height-3 WTD balanced trees from a base path $P_6$; these together imply polynomial-time recognition for WTD trees. The central results include a descriptive criterion for balanced WTD trees based on height and neighborhood constraints, a reduction of WTD testing for arbitrary trees to the WTD-ness of their interior graphs, and a constructive framework that generates all height-3 WTD balanced trees. Collectively, these contributions deepen structural understanding of WTD trees and enable efficient verification and systematic construction of such 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.

Figures (20)

• Figure 1: Trees with 2-colorings
• Figure 2: Graph $Y$ with a 2-coloring
• Figure 3: Branch and radar example on $G$
• Figure 4: Tree $T$ with vertices labeled and colored
• Figure 5: Tree $T$ with branches (there are branches growing from all $s_i$'s like $s_m$)

Theorems & Definitions (69)

• Definition 2.2
• Definition 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Definition 2.6
• Lemma 2.7
• proof
• Definition 2.8