Metric dimensions of bicyclic graphs

Abstract

The distance d(va, vb) between two vertices of a simple connected graph G is the length of the shortest path between va and vb. Vertices va, vb of G are considered to be resolved by a vertex v if d(va, v) 6= d(vb, v). An ordered set W = fv1, v2, v3, . . . , vsg V(G) is said to be a resolving set for G, if for any va, vb 2 V(G), 9 vi 2 W 3 d(va, vi) 6= d(vb, vi). The representation of vertex v with respect to W is denoted by r(vjW) and is an s-vector(s-tuple) (d(v, v1), d(v, v2), d(v, v3), . . . , d(v, vs)). Using representation r(vjW), we can say that W is a resolving set if, for any two vertices va, vb 2 V(G), we have r(vajW) 6= r(vbjW). A minimal resolving set is termed a metric basis for G. The cardinality of the metric basis set is called the metric dimension of G, represented by dim(G). In this article, we study the metric dimension of two types of bicyclic graphs. The obtained results prove that they have constant metric dimension.