The Geodetic Fault Tolerant Domination Number of a Graph

Abstract

For a connected graph G = (V,E), a set F ⊆ V of vertices in G is called dominating set if every vertex not in F has at least one neighbor in F. A dominating set F ⊆ V is called fault tolerant dominating set if F − {v} is dominating set for every v ∈ F. A fault tolerant dominating set is said to be geodetic fault tolerant dominating set if I[F] = V . The minimum cardinality of a geodetic fault tolerant dominating set is called geodetic fault tolerant domination number and is denoted by γgft(G). The minimum geodetic fault tolerant dominating set is denoted by γgft-set. The geodetic fault tolerant domination number of certain classes of graphs are determined. Some general properties satisfied by this concept are studied. It is shown that for every positive integer 2 < a ≤ b there is a connected graph G such that γ(G) = a, γg(G) = b and γgft(G) = a + b − 2, where γ(G) and γg(G) are the domination number and geodetic domination number of G respectively.