Secondary and Internal Distances of Sets in Graphs

Abstract

For any given type of a set of vertices in a connected graph G = (V, E), we seek to determine the smallest integers (x, y: z) such that all minimal (or maximal) sets S of the given type, where |V| > |S| ≥ 2, have the property that every vertex v ∊ V – S is within distance at most x to a vertex u ∊ S (shortest distance), and within distance at most y to a second vertex w ∊ S (second shortest distance). We also seek to determine the smallest integer z such that every vertex u ∊ S is within distance at most z to a closest neighbor w ∊ S (the internal distance). A dominating set S ⊆ V in a graph G is a set having the property that every vertex v ∊ V – S is within distance 1 to some vertex in S, or equivalently, whose shortest distance x = 1. In this paper we determine the secondary distances y and internal distances z for 31 types of sets in graphs, whose shortest distance x = 1.