The Lower and Upper Forcing Geodetic Numbers of Block-Cactus Graphs

Abstract

A vertex set D in graph G is called a geodetic set if all vertices of G are lying on some shortest u − v path of G, where u, v 2 D. The geodetic number of a graph G is the minimum cardinality among all geodetic sets. A subset S of a geodetic set D is called a forcing subset of D if D is the unique geodetic set containing S. The forcing geodetic number of D is the minimum cardinality of a forcing subset of D, and the lower and upper forcing geodetic numbers of a graph G are the minimum and maximum forcing geodetic numbers, respectively, among all geodetic sets of G. In this paper, we find out the lower and upper forcing geodetic numbers of block-cactus graphs.