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arXiv · 1711.01951

Locating domination in bipartite graphs and their complements

Carmen Hernando·Mercè Mora·Ignacio M. Pelayo
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Abstract

A set $S$ of vertices of a graph $G$ is \emph{distinguishing} if the sets of neighbors in $S$ for every pair of vertices not in $S$ are distinct. A \emph{locating-dominating set} of $G$ is a dominating distinguishing set. The \emph{location-domination number} of $G$, $\lambda(G)$, is the minimum cardinality of a locating-dominating set. In this work we study relationships between $\lambda({G})$ and $\lambda (\overline{G})$ for bipartite graphs. The main result is the characterization of all connected bipartite graphs $G$ satisfying $\lambda (\overline{G})=\lambda({G})+1$. To this aim, we define an edge-labeled graph $G^S$ associated with a distinguishing set $S$ that turns out to be very helpful.

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Carmen Hernando, Mercè Mora, Ignacio M. Pelayo. 2017-11-03. Locating domination in bipartite graphs and their complements. https://arxiv.org/abs/1711.01951

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