On the vertex position number of graphs

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 ⊆ V (G) is an x-position set if for any y ∈ S the shortest 2 Thankachy, Chandran, Tuite, Thomas, Di Stefano and Erskinex, y-paths in G contain no point of S \ {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.

Lower general position sets in graphs

“…• As discussed in Section 1, there are several noteworthy variations of the general position number in the literature, including the mutual visibility number [15], d-position sets [26], vertex position numbers [37], Steiner position numbers [24], edge general position numbers [30], mobile position sets [23], etc. We suggest studying lower versions of these parameters.…”

“…Several variations of the problem have been considered, including using the Steiner distance instead of the regular graph distance [24], or confining attention to shortest paths of bounded length [26]. Games involving general position sets have also been treated in [10] and [25], a dynamic variant of the problem was considered in [23] and a local version of general position sets was studied in [37].…”