On the local average order of dominating sets

@misc{chen2026local,
  title = {{On the local average order of dominating sets}},
  author = {Tingyun Chen and Weihua He and Hong-Jian Lai and Jianping Li},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11446},
  eprint = {2610.11446},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11446},
}

TY  - UNPB
TI  - On the local average order of dominating sets
AU  - Tingyun Chen
AU  - Weihua He
AU  - Hong-Jian Lai
AU  - Jianping Li
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.11446
UR  - https://arxiv.org/abs/2610.11446
AB  - The (global) average order of dominating sets of a graph is the average number of vertices of its dominating sets. Analogously, the local average order of dominating sets is the average number of vertices of its dominating sets containing a fixed vertex. In this paper, we show that the local average order of dominating sets of a graph with n vertices is at least (n+1)/2, with equality if and only if the degree of the fixed vertex is n−1. Furthermore, for a graph on n vertices without isolated vertices, we show that (5n−1)/6 is an upper bound for the local average order of dominating sets. Additionally, we give a proof of an exact formula for the local average order of dominating sets when the degree of the fixed vertex is n−2, and determine an upper bound for the local average order of dominating sets when the fixed vertex is an l-stem (l ≥ 2).
ER  -