On Gated Sets in Graphs

Abstract

A subset $K$ of vertices of a graph $G$ is gated if for every vertex $x \in V(G)$ there exists a gate $v \in K$ which is on a shortest path between $x$ and any vertex $u$ of $K$. We give a characterization of gated sets in an arbitrary graph $G$ and several necessary conditions. This characterization yields very nice results in the case of weakly modular graphs, which are also presented. We also show that the trees are precisely the graphs which present a convex geometry with respect to the gated convexity.