On randomly k-dimensional graphs

Abstract

For an ordered set W=w1,w2,⋯,wk of vertices and a vertex v in a connected graph G, the ordered k-vector r(v|W):=(d(v,w1),d(v,w2),⋯,d(v,wk)) is called the (metric) representation of v with respect to W, where d(x,y) is the distance between the vertices x and y. The set W is called a resolving set for G if distinct vertices of G have distinct representations with respect to W. A resolving set for G with minimum cardinality is called a basis of G and its cardinality is the metric dimension of G. A connected graph G is called a randomly k-dimensional graph if each k-set of vertices of G is a basis of G. In this work, we study randomly k-dimensional graphs and provide some properties of these graphs. © 2011 Elsevier Ltd. All rights reserved.