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On the efficiency index of a graph

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Abstract

A graph \(G\) has an efficient dominating set \(D \subseteq V(G)\) if \(D\) dominates every vertex exactly once. In this paper we introduce the study of the family \({S_k}\) of graphs for which every \(G-S\) is efficiently dominatable for \(0 \le |S|\le k\). Assuming that \(G\) is efficiently dominatable, the efficiency index is the largest value k for which \(G\) is in \(S_k\). A graph \(G\) will be called super-efficient if every induced subgraph is efficiently dominatable. We give some characterizations for trees, grids, cylinders and torii to be super-efficient.

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Correspondence to Rommel Barbosa.

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Barbosa, R., Slater, P. On the efficiency index of a graph. J Comb Optim 31, 1134–1141 (2016). https://doi.org/10.1007/s10878-014-9814-1

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  • DOI: https://doi.org/10.1007/s10878-014-9814-1

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