The hull and geodetic numbers of orientations of graphs

Abstract

For an oriented graph D , let I D [ u , v ] denote the set of all vertices lying on a u – v geodesic or a v – u geodesic. For S ⊆ V ( D ) , let I D [ S ] denote the union of all I D [ u , v ] for all u , v ∈ S . Let [ S ] D denote the smallest convex set containing S . The geodetic number g ( D ) of an oriented graph D is the minimum cardinality of a set S with I D [ S ] = V ( D ) and the hull number h ( D ) of an oriented graph D is the minimum cardinality of a set S with [ S ] D = V ( D ) . For a connected graph G , let O ( G ) be the set of all orientations of G , define g − ( G ) = min { g ( D ) : D ∈ O ( G ) } , g + ( G ) = max { g ( D ) : D ∈ O ( G ) } , h − ( G ) = min { h ( D ) : D ∈ O ( G ) } , and h + ( G ) = max { h ( D ) : D ∈ O ( G ) } . By the above definitions, h − ( G ) ≤ g − ( G ) and h + ( G ) ≤ g + ( G ) . In the paper, we prove that g − ( G ) < h + ( G ) for a connected graph G of order at least 3, and for any nonnegative integers a and b , there exists a connected graph G such that g − ( G ) − h − ( G ) = a and g + ( G ) − h + ( G ) = b . These results answer a problem of Farrugia in [A. Farrugia, Orientable convexity, geodetic and hull numbers in graphs, Discrete Appl. Math. 148 (2005) 256–262].