The forcing hull and forcing geodetic numbers of graphs
For every pair of vertices u,v u , v in a graph, a uv u  v geodesic is a shortest path from u u to v v . For a graph G G , let IG[u,v] I G [ u , v ] denote the set of all vertices lying on a uv u  v geodesic. Let S?V(G) S ? V ( G ) and IG[S] I G [ S ] denote the union of all IG[u,v] I G [ u , v ] for all u,v?S u , v ? S . A subset S?V(G) S ? V ( G ) is a convex set of G G if IG[S]=S I G [ S ] = S . A convex hull [S]G [ S ] G of S S is a minimum convex set containing S S . A subset S S of V(G) V ( G ) is a hull set of G G if [S]G=V(G) [ S ] G = V ( G ) . The hull number h(G) h ( G ) of a graph G G is the minimum cardinality of a hull set in G G . A subset S S of V(G) V ( G ) is a geodetic set if IG[S]=V(G) I G [ S ] = V ( G ) . The geodetic number g(G) g ( G ) of a graph G G is the minimum cardinality of a geodetic set in G G . A subset F?V(G) F ? V ( G ) is called a forcing hull (or geodetic) subset of G G if there exists a unique minimum hull (or geodetic) set containing F F . The cardinality of a minimum forcing hull subset in G G is called the forcing hull number fh(G) f h ( G ) of G G and the cardinality of a minimum forcing geodetic subset in G G is called the forcing geodetic number fg(G) f g ( G ) of G G . In the paper, we construct some 2-connected graph G G with (fh(G),fg(G))=(0,0),(1,0) ( f h ( G ) , f g ( G ) ) = ( 0 , 0 ) , ( 1 , 0 ) , or (0,1) ( 0 , 1 ) , and prove that, for any nonnegative integers a a , b b , and c c with a+b?2 a + b ? 2 , there exists a 2-connected graph G G with (fh(G),fg(G),h(G),g(G))=(a,b,a+b+c,a+2b+c) ( f h ( G ) , f g ( G ) , h ( G ) , g ( G ) ) = ( a , b , a + b + c , a + 2 b + c ) or (a,2a+b,a+b+c,2a+2b+c) ( 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) 8194].
forcing geodetic number geodetic number geodesic hull number forcing hull number