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arXiv · 2602.13956

Characterization of Well-Totally Dominated Trees

Jounglag Lim·James Gossell·Keri Ann Sather-Wagstaff·Devin Adams·Suzanna Castro-Tarabulsi·Aayahna Herbert·Vi Anh Nguyen·Yifan Qian·Matthew Schaller·Zoe Zhou·Yuyang Zhuo
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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.

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Jounglag Lim, James Gossell, Keri Ann Sather-Wagstaff, Devin Adams, Suzanna Castro-Tarabulsi, Aayahna Herbert, Vi Anh Nguyen, Yifan Qian, Matthew Schaller, Zoe Zhou, Yuyang Zhuo. 2026-02-15. Characterization of Well-Totally Dominated Trees. https://arxiv.org/abs/2602.13956

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