The geodetic domination number of comb product graphs

Dimas Agus Fahrudin (Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Indonesia)
Suhadi Wido Saputro (Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Indonesia)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 8, No 2 (2020): Electronic Journal of Graph Theory and Applications

A subset S of vertices in graph G is called a geodetic set if every vertex in V(G) \ S lies on a shortest path between two vertices in S. A subset S of vertices in G is called a dominating set if every vertex in V(G) \  S is adjacent to a vertex in S. The set S is called a geodetic dominating set if S is both geodetic and dominating sets. The geodetic domination number of G, denoted by γg(G), is the minimum cardinality of geodetic domination sets in G. The comb product of connected graphs G and H at vertex o ∈ V(H), denoted by  G ∇o H, is a graph obtained by taking one copy of G and |V(G)| copies of H and identifying the ith copy of H at the vertex o to the ith vertex of G. In this paper, we determine an exact value of γg(G ∇o H) for any connected graphs G and H.