Abstract:A dominating set is called $k$-limited if every vertex in the set has at most $k$ neighbors outside it. The minimum cardinality of a $k$-limited dominating set is the $k$-limited domination number, denoted by $\gamma_k^{\mathrm{L}}(G)$. 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 $\gamma_k^{\mathrm{L}}(G\square H)$, show that both are sharp, and derive exact values for several natural families of graph products. Among others, we obtain exact results for rook graphs, Cartesian products of $k$-coronas, certain grid graphs, and several cases involving prisms and hypercubes.
From: Aleksandra Tepeh [view email]
[v1]
Sun, 21 Jun 2026 10:31:28 UTC (20 KB)
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