On Weakly Connected Closed Geodetic Domination in Graphs Under Some Binary Operations

Let G be a simple connected graph. For S subset of V(G), the weakly connected closed geodetic dominating set S of G is a geodetic closure I-G[S] which is between S and is the set of all vertices on geodesics (shortest path) between two vertices of S. We select vertices of G sequentially as follows: Select a vertex vi and let S-1 = {v(1)}. Select a vertex v(2) not equal v(1) and let S-2 = {v(1), v(2)}. Then successively select vertex v(i) is not an element of I-G [Si-1] and let S-i = {v(1), v(2), ..., v(i)} for i = 1,2, ..., k until we select a vertex v(k) in the given manner that yields I-G[S-k] = V(G). Also, the subgraph weakly induced < S >(w) by S is connected where < S >(w) = < N[S], E-w > with E-w = {u, v is an element of E(G): u is an element of S or v is an element of S} and S is a dominating set of G. The minimum cardinality of weakly connected closed geodetic dominating set of G is denoted by gamma(wcg)(G). In this paper, the authors show and investigate the concept weakly connected closed geodetic dominating sets of some graphs and the join, corona, and Cartesian product of two graphs are characterized. The weakly connected closed geodetic domination numbers of these graphs are determined. Also, some relationships between weakly connected closed geodetic dominating set, weakly connected closed geodetic set, geodetic dominating set, and geodetic connected dominating set are established.