The geodetic domination number of comb product graphs
Dimas Agus Fahrudin, Suhadi Wido Saputro · Electronic Journal of Graph Theory and Applications · 2020
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 γ g ( 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 i th copy of H at the vertex o to the i th vertex of G . In this paper, we determine an exact value of γ g ( G ∇ o H ) for any connected graphs G and H .