Polynomial time algorithm for computing a minimum geodetic set in outerplanar graphs

Abstract

Given a graph G and a pair of vertices u,v the interval IG[u,v] is the set of all vertices that are in some shortest path between u and v. Given a subset X of vertices of G, the interval IG[X] of X, is the union of the intervals for all pairs of vertices in X and we say that X is geodetic if its interval do coincide with the set of vertices in the graph. A minimum geodetic set is a minimum cardinality geodetic set of G. The problem of computing a minimum geodetic set is known to be NP-Hard for general graphs but is known to be polynomially solvable for maximal outerplanar graphs. In this paper we show a polynomial time algorithm for finding a minimum geodetic set in general outerplanar graphs.