On the geodetic number of a graph
Gary Chartrand, Frank Harary, Ping Zhang · Networks · 2001
Abstract For two vertices u and v of a graph G, the set I(u, v) consists of all vertices lying on some u − v geodesic 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). It is shown that if G is a graph of order n and diameter d then g(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 of radius r, diameter d, and g(G) = k. Also, for integers n, d, and k with 2 ≤ d < n, 2 ≤ k < n, and n − d − k + 1 ≥ 0, there exists a graph G of order n, diameter d, and g(G) = k. It is shown, for every nontrivial connected graph G, that g(G) ≤ g(G × K2). A sufficient condition for the equality of g(G) and g(G × K2) is presented. A graph F is a minimum geodetic subgraph if there exists a graph G containing F as an induced subgraph such that V(F) is a minimum geodetic set for G. Minimum geodetic subgraphs are characterized. © 2002 John Wiley & Sons, Inc.