The near geodetic number of a graph

Abstract

The interval I(u, v) for any two vertices consists of all those vertices lying on a u – v geodesic (shortest path) in G. If S is a set of vertices of G, then I(S) is the union of all sets I(u, v) for u, v ∈ S. The geodetic number g(G) is the minimum cardinality among the subsets S of V(G) with I(S) = V(G). A set S ⊇ V(G) is a near geodetic set if for every v in V(G) – S, there exists some x, y in S with |I(x, y) ∩ N(v)|≥ 2. The near geodetic number gnear(G) is the minimum cardinality of a near geodetic set in G. In this paper, we proved that if G is a graph of order n and clique number w (G), then gnear(G) ≤ n − ω(G) + 2 and this bound is sharp. Further we proved that any positive integers a, b and n, with 2 ≤ a < b ≤ n − 2, there exists a connected graph G of order n with g(G) = a and gnear(G) = b. Also we shown that, for positive integers n,k,l with n − k − l + 2 ≥ 0, 2 ≤ k < n − 1 and 2 ≤ l ≤ n, there exist a connected graph G of order n with clique number ω (G) = 1 and gnear(G) = K.