We show that the geodetic number of proper interval graphs can be computed in polynomial time. This problem is NP-hard on chordal graphs and on bipartite weakly chordal graphs. Only an upper bound on the geodetic number of proper interval graphs has been known prior to

For a connected graph G = (V,E), a set W ⊆ V is called a Steiner set of G if every vertex of G is contained in a Steiner W -tree of G. The Steiner number s(G) of G is the minimum cardinality of its Steiner sets and any Steiner set of cardinality s(G) is a minimum Steiner set of G. For a minimum Steiner set W of G, a subset T ⊆ W is called a forcing subset for W if W is the unique minimum Steine...

A set S ⊆ V (G) is called a geodetic if every vertex of G lies on shortest u-v path for some u, v ∈ S, the minimum cardinality among all sets number and denoted by . C chromatic contains vertices different colors in G, geo-chromatic Sc both set. The G. In this paper, we determine 2-cartesian product standard graphs like complete graphs, cycles paths.

In the geodetic convexity, a set of vertices S graph G is convex if all belonging to any shortest path between two lie in . The hull H ( ) smallest containing If = V ), then cardinality h minimum number complementary prism GḠ arises from disjoint union and Ḡ by adding edges perfect matching corresponding A autoconnected both are connected. Motivated previous work, we study for prisms graphs. Wh...

A set S of vertices of a graph G is a geodetic set if every vertex of G lies in an interval between two vertices from S. The size of a minimum geodetic set in G is the geodetic number g(G) of G. We find that the geodetic number of the lexicographic product G ∘H for non-complete graphs H lies between 2 and 3g(G). We characterize the graphs G and H for which g(G ∘H) = 2, as well as the lexicograp...