Characterization of Well-Totally Dominated Trees
Combinatorics
2026-02-17 v1
Abstract
Let be a graph with no isolated vertices. A set of vertices is a total dominating set (TDS) if every vertex in is adjacent to at least one vertex in . We say 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.
Keywords
Cite
@article{arxiv.2602.13956,
title = {Characterization of Well-Totally Dominated Trees},
author = {Jounglag Lim and James Gossell and Keri Ann Sather-Wagstaff and Devin Adams and Suzanna Castro-Tarabulsi and Aayahna Herbert and Vi Anh Nguyen and Yifan Qian and Matthew Schaller and Zoe Zhou and Yuyang Zhuo},
journal= {arXiv preprint arXiv:2602.13956},
year = {2026}
}
Comments
This paper, together with the revised version of arXiv:2510.19707 (v4), represents a split of the original v1 of arXiv:2510.19707