Closed Geodetic Hop Domination in Graphs

Let G be a simple, undirected and connected graph. A subset S ⊆ V (G) is a geodetic cover of G if IGS = V (G), where IGS is the set of all vertices of G lying on any geodesic between two vertices in S. A geodetic cover S of G is a closed geodetic cover if the vertices in S are sequentially selected as follows: Select a vertex v1 and let S1 = v1. If G is nontrivial, select a vertex v2 ̸= v1 and let S2 = v1, v2. Where possible, for i ≥ 3, successively select vertex vi ∈/ IGSi−1 and let Si = v1, v2,. . . , vi. Then there exists a positive integer k such that Sk = S. A geodetic cover S of G is a geodetic hop dominating set if every vertex in V (G) \ S is of distance2 from a vertex in S. A geodetic hop dominating set S is a closed geodetic hop dominating set if S is a closed geodetic cover of G. The minimum cardinality of a (closed) geodetic hop dominating set of G is the (closed) geodetic hop domination number of G. This study initiates the study of the closed geodetic hop domination. First, it characterizes all graphs G of order n whose closed geodetic hop domination numbers are 2 or n, and determines the closed geodetic hop domination number of paths, cycles and multigraphs. Next, it shows that any positive integers a and b with 2 ≤ a ≤ b are realizable as the closed geodetic number and closed geodetic hop domination number of a connected graph. Also, every positive integer n, m and k with 4 ≤ m ≤ k and 2k−m+2 ≤ n are realizable as the order, geodetic hop domination number and closed geodetic hop domination number, respectively of a connected graph. Furthermore, the study characterizes the closed geodetic hop dominating sets of graphs resulting from the join, corona and edge corona of graphs.