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arXiv · 1809.08630

Bounds on the nonnegative signed domination number of graphs

Doost Ali Mojdeh·Babak Samadi·Lutz Volkmann
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Abstract

The aim of this work is to investigate the nonnegative signed domination number $\gamma^{NN}_s$ with emphasis on regular, ($r+1$)-clique-free graphs and trees. We give lower and upper bounds on $\gamma^{NN}_s$ for regular graphs and prove that $n/3$ is the best possible upper bound on this parameter for a cubic graph of order $n$, specifically. As an application of the classic theorem of Tur\'{a}n we bound $\gamma^{NN}_s(G)$ from below, for an ($r+1$)-clique-free graph $G$ and characterize all such graphs for which the equality holds, which corrects and generalizes a result for bipartite graphs in [Electron. J. Graph Theory Appl. 4 (2) (2016), 231--237], simultaneously. Also, we bound $\gamma^{NN}_s(T)$ for a tree $T$ from above and below and characterize all trees attaining the bounds.

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Doost Ali Mojdeh, Babak Samadi, Lutz Volkmann. 2018-09-23. Bounds on the nonnegative signed domination number of graphs. https://arxiv.org/abs/1809.08630

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