Computing a metric basis of a 2-connected bipartite distance-hereditary graph

A vertex x of a connected graph G resolves two distinct vertices u and v in V(G) if the distance between u and x differs from the distance between v and x. A subset X of V(G) resolves two distinct vertices u and v in G if there exists a vertex x in X that resolves u and v; X is a metric generator of G if, for every pair of distinct vertices u and v of G, X resolves u and v and is a metric basis of G if X is a metric generator of G with minimum cardinality. The metric dimension of G is the cardinality of a metric basis of G. The problem of finding the metric dimension of an arbitrary graph is NP-hard. In this paper we show that the problem is solvable in polynomial time for the class of 2-connected bipartite distance-hereditary graphs by providing an algorithm that computes a metric basis of a 2-connected bipartite distance-hereditary graph in O(|V(G)|2|E(G)|) time.