For a connected graph G of order n, a set S ⊆ V (G) is a geodetic set of G if each vertex v ∈ V (G) lies on a x-y geodesic for some elements x and y in S. The minimum cardinality of a geodetic set of G is defined as the geodetic number of G, denoted by g(G). A geodetic set of cardinality g(G) is called a g-set of G. A set S of vertices of a connected graph G is an open geodetic set of G if for ...

For a connected graph G = (V,E), a set S ⊆ E is called an edge-to-vertex geodetic set of G if every vertex of G is either incident with an edge of S or lies on a geodesic joining some pair of edges of S. The minimum cardinality of an edge-to-vertex geodetic set of G is gev(G). Any edge-to-vertex geodetic set of cardinality gev(G) is called an edge-to-vertex geodetic basis of G. A subset T ⊆ S i...

For two vertices u and v of a graph G, the closed interval I[u, v] consists of u, v, and all vertices lying in some u − v geodesic in G. If S is a set of vertices of G, then I[S] is the union of all sets I[u, v] for u, v ∈ S. If I[S] = V (G), then S is a geodetic set for G. The geodetic number g(G) is the minimum cardinality of a geodetic set. A set S of vertices in a graph G is uniform if the ...

We study convexity properties of graphs. In this paper we present a linear-time algorithm for the geodetic number in tree-cographs. Settling a 10-year-old conjecture, we prove that the Steiner number is at least the geodetic number in AT-free graphs. Computing a maximal and proper monophonic set in AT-free graphs is NP-complete. We present polynomial algorithms for the monophonic number in perm...

A set of vertices S in a graph is called geodetic if every vertex of this graph lies on some shortest path between two vertices from S. In this paper, minimum geodetic sets in median graphs are studied with respect to the operation of peripheral expansion. Along the way geodetic sets of median prisms are considered and median graphs that possess a geodetic set of size two are characterized. © 2...

Given a simple connected graph G = (V, E) the geodetic closure I[S] ⊂ V of a subset S of V is the union of all sets of nodes lying on some geodesic (or shortest path) joining a pair of nodes vk, vl ∈ S. The geodetic number, denoted by g(G), is the smallest cardinality of a node set S∗ such that I[S∗] = V . In “The geodetic number of a graph”, Mathematical and Computer Modelling 17 (June 1993) 8...