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An inequality that relates the size of a bipartite graph with its order and restrained domination number

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Abstract

Let \(G=(V,E)\) be a graph. A set \(S\subseteq V\) is a restrained dominating set if every vertex in \(V-S\) is adjacent to a vertex in \(S\) and to a vertex in \(V-S\). The restrained domination number of \(G\), denoted \(\gamma _{r}(G)\), is the smallest cardinality of a restrained dominating set of \(G\). Consider a bipartite graph \(G\) of order \(n\ge 4,\) and let \(k\in \{2,3,...,n-2\}.\) In this paper we will show that if \(\gamma _{r}(G)=k\), then \(m\le ((n-k)(n-k+6)+4k-8)/4\). We will also show that this bound is best possible.

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Correspondence to Ernst J. Joubert.

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Joubert, E.J. An inequality that relates the size of a bipartite graph with its order and restrained domination number. J Comb Optim 31, 44–51 (2016). https://doi.org/10.1007/s10878-014-9709-1

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