THE LINEAR GEODETIC NUMBER OF A GRAPH

For a connected graph G of order n, an ordered set S = {u(1), u(2),..., u(k)} of vertices in G is a linear geodetic set of G if for each vertex x in G, there exists an index i, 1 <= i < k such that x lies on a u(i) - u(i+1) geodesic on G, and a linear geodetic set of minimum cardinality is the linear geodetic number gl(G). The linear geodetic numbers of certain standard graphs are obtained. It is shown that if G is a graph of order n and diameter d, then gl(G) <= n - d + 1 and this bound is sharp. For positive integers r, d and k >= 2 with r < d <= 2r, there exists a connected graph G with rad G = r, diamG = d and gl(G) = k. Also, for integers n, d and k with 2 <= d < n, 2 <= k <= n - d + 1, there exists a connected graph G of order n, diameter d and gl(G) = k. We characterize connected graphs G of order n with gl(G) = n and gl(G) = n - 1. It is shown that for each pair a, b of integers with 3 <= a <= b, there is a connected graph G with g(G) = a and gl(G) = b. We also discuss how the linear geodetic number of a graph is affected by adding a pendent edge to the graph.