The forcing hull and forcing geodetic numbers of graphs

Abstract

For every pair of vertices u , v in a graph, a u – v geodesic is a shortest path from u to v . For a graph G , let I G [ u , v ] denote the set of all vertices lying on a u – v geodesic. Let S ⊆ V ( G ) and I G [ S ] denote the union of all I G [ u , v ] for all u , v ∈ S . A subset S ⊆ V ( G ) is a convex set of G if I G [ S ] = S . A convex hull [ S ] G of S is a minimum convex set containing S . A subset S of V ( G ) is a hull set of G if [ S ] G = V ( G ) . The hull number h ( G ) of a graph G is the minimum cardinality of a hull set in G . A subset S of V ( G ) is a geodetic set if I G [ S ] = V ( G ) . The geodetic number g ( G ) of a graph G is the minimum cardinality of a geodetic set in G . A subset F ⊆ V ( G ) is called a forcing hull (or geodetic) subset of G if there exists a unique minimum hull (or geodetic) set containing F . The cardinality of a minimum forcing hull subset in G is called the forcing hull number f h ( G ) of G and the cardinality of a minimum forcing geodetic subset in G is called the forcing geodetic number f g ( G ) of G . In the paper, we construct some 2-connected graph G with ( f h ( G ) , f g ( G ) ) = ( 0 , 0 ) , ( 1 , 0 ) , or ( 0 , 1 ) , and prove that, for any nonnegative integers a , b , and c with a + b ≥ 2 , there exists a 2-connected graph G with ( f h ( G ) , f g ( G ) , h ( G ) , g ( G ) ) = ( a , b , a + b + c , a + 2 b + c ) or ( a , 2 a + b , a + b + c , 2 a + 2 b + c ) . These results confirm a conjecture of Chartrand and Zhang proposed in [G. Chartrand, P. Zhang, The forcing hull number of a graph, J. Combin. Math. Combin. Comput. 36 (2001) 81–94].