• ON GEODETIC SETS AND POLYNOMIALS OF CENTIPEDES

Abstract

Let G = (V,E) be a simple graph. A set of vertices S of a graph G is geodetic, if every vertex of G lies on a shortest path between two vertices in S. The geodetic number of G is the minimum cardinality of all geodetic sets of G, and is denoted by g (G). In (8), the concept of geodetic polynomial is defined as where is the number of geodetic sets of cardinality i, and be the graph obtained by appending a single pendant edge to each vertex of graph G. We call a centipede, where is a path with n vertices. In this paper, we obtain the geodetic sets and polynomials of the centipedes. Also, we study some properties of geodetic sets and the coefficients of the polynomials. It is also derived that the geodetic polynomial of the centipede is .