The Geodesic Irredundant Sets in Graphs

For two vertices u and v of a connected graph G, the set I[u,v] consists of all those vertices lying on u - v geodesics in G. Given a set S of vertices of G, the union of all sets I[u,v] for u,v ∈ S is denoted by I[S]. A convex set S satisfies I[S] = S. The convex hull [S] of S is the smallest convex set containing S. The hull number h(G) is the minimum cardinality among the subsets S of V with [S] = V. In this paper, we introduce and study the geodesic irredundant number of a graph. A set S of vertices of G is a geodesic irredundant set if u ∉ I[S - {u}] for all u ∈ S and the maximum cardinality of a geodesic irredudant set is its irredundant number gir(G) of G. We determine the irredundant number of certain standard classes of graphs. Certain general properties of these concepts are studied. We characterize the classes of graphs of order n for which gir(G) = 2 or gir(G) = n or gir(G) = n - 1, respectvely. We prove that for any integers a and b with 2 ≤ a ≤ b, there exists a connected graph G such that h(g) = a and gir(G) = b. A graph H is called a maximum irredundant subgraph if there exists a graph G containing H as induced subgraph such that V (H) is a maximum irredundant set in G. We characterize the class of maximum irredundant subgraphs.