The -forcing geodetic graphs

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, and for S ⊆ V ( G ) , let I G [ S ] denote the union of all I G [ u , v ] for all u , v ∈ S . A set S ⊆ 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 subset of G if there exists a unique minimum geodetic set containing F . A forcing subset F is critical if every proper subset of F is not a forcing subset. The cardinality of a minimum critical forcing subset in G is called the forcing geodetic number f ( G ) of G and the cardinality of a maximum critical forcing subset in G is called the upper forcing geodetic number f + ( G ) of G . If G is a graph with f ( G ) = 0 , then G has a unique minimum geodetic set; that is, f + ( G ) = 0 . In the paper, we prove that, for any nonnegative integers a , b and c with 1 ≤ a ≤ b ≤ c − 2 or 4 ≤ a + 2 ≤ b ≤ c , there exists a connected graph G with f ( G ) = a , f + ( G ) = b , and g ( G ) = c . This result solves a problem of Zhang [P. Zhang, The upper forcing geodetic number of a graph, Ars Combin. 62 (2002) 3–15].