On connected closed geodetic numbers of some graphs

Abstract

This study deals primarily with the connected closed geodetic numbers of some graphs, a closed geodetic closure invariant introduced by Buckley and Harary [2]. For S ⊆ V (G), where G is a connected graph, the geodetic closure I G [S] of S is the set of all vertices lying on some u-v geodesic where u and v are in S. In this paper, select vertices of G sequentially as follows: Select a vertex v 1 and let S 1 = {v 1 }. Select a vertex v 2 = v 1 and let S 2 = {v 1 , v 2 }, then determine I G [S 2 ]. If I G [S 2 ] = V (G), then successively select vertex v i / ∈ I G [S i−1 ] and let S i = {v 1 , v 2 , ..., v i } for i = 1, 2, ..., k. Then determine I G [S i ]. The connected closed geodetic number of a graph, denoted by ccgn(G), is defined to be the smallest k whose selection of v k in the given manner yields I G [S k ] = V (G), where S is connected. In this paper, the connected closed geodetic numbers of some graphs and the join of some connected graphs G and H were determined.

On connected closed geodetic number of the join of graphs

“…Amanodin in [1] also proved that the ccgn(P n + K p ) = n + 1 2 + 1, for any p ≥ 1 and n ≥ 3. In particular, we study the graph P 5 + K 4 which is given in the next example.…”

“…The study on closed geodetic numbers leads the researchers to study closely on connected closed geodetic numbers. Amanodin in [1] studied on connected closed geodetic numbers of some special graphs. The results in [2] and [1] motivate the researchers to investigate further the behavior of S ⊆ V (G), where S ∈ C * (G) and S is connected, for the join of two graphs.…”