On the vertex position number of graphs

Maya G. S. Thankachy, S. V. Ullas Chandran, James Tuite, Elias John Thomas, Gabriele Di Stefano, Grahame Erskine

Abstract: In this paper we generalise the notion of visibility from a point in an integer lattice to the setting of graph theory. For a vertex $x$ of a graph $G$, we say that a set $S \subseteq V(G)$ is an $x$-position set if for any $y \in S$ the shortest $x,y$-paths in $G$ contain no point of $S\setminus \{ y\}$. We investigate the largest and smallest orders of maximum $x$-position sets in graphs, determining these numbers for common classes of graphs and giving bounds in terms of the girth, vertex degrees, diameter and radius. Finally we discuss the complexity of finding maximum vertex position sets in graphs.