Geodetic numbers of tensor product and lexicographic product of graphs

Abstract

A shortest [Formula: see text]-[Formula: see text] path between two vertices u and v of a graph G is a [Formula: see text]-[Formula: see text] geodesic of G. Let I[u, v] denote the set of all internal vertices lying on some [Formula: see text]-[Formula: see text] geodesic of G. For a nonempty subset S of [Formula: see text], let [Formula: see text]. If [Formula: see text], then S is a geodetic set of G. The cardinality of a minimum geodetic set of G is the geodetic number of G and it is denoted by [Formula: see text] In this paper, the exact geodetic numbers of the product graphs [Formula: see text] and [Formula: see text] are obtained, where T is a tree, [Formula: see text] denotes the complement of the complete graph [Formula: see text] and, [Formula: see text] and [Formula: see text] denote the tensor product and lexicographic product $($also called the wreath product$)$ of graphs, respectively.

Keywords

• Distance
• geodesic
• geodetic number
• tensor product
• lexicographic product
• 05C12