Year: 2026
Type: preprint
Abstract: A dominating set is called $k$-limited if every vertex in the set has at most $k$ neighbors outside it. The minimum cardinality among all $k$-limited dominating sets of a graph $G$ is the $k$-limited domination number, denoted by $γ_k^{\mathrm L}(G)$. We prove that, for every fixed integer $k\ge 2$, deciding whether a graph admits a $k$-limited dominating set of size at most $\ell$ is $\mathsf{NP}$-complete. In addition, a systematic study of $k$-limited domination in Cartesian products is initiated. In particular, we establish general lower and upper bounds for $γ_k^{\mathrm L}(G\square H)$, show that both are sharp, and derive exact values for several natural families of graph products.
Source: arXiv (Cornell University)
Author: Tepeh, Aleksandra
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7165635185