The Resolving Graph of Amalgamation of Cycles
Hazrul Iswadi, Edy Tri Baskoro, A.N.M. Salman, Rinovia Simanjuntak · Ubaya Repository (University of Surabaya) · 2010
For an ordered set W = {w_1,w_2,...,w_k} of vertices and a vertex v in a connected graph G, the representation of v with respect to W is the ordered k-tuple r(v|W) = (d(v,w_1),d(v,w_2),...,d(v,w_k)) where d(x,y) represents the distance between the vertices x and y. The set W is called a resolving set for G if every vertex of G has a distinct representation. A resolving set containing a minimum number of vertices is called a basis for G. The dimension of G, denoted by dim(G), is the number of vertices in a basis of G. A resolving set W of G is connected if the subgraph induced by W is a nontrivial connected subgraph of G. The connected resolving number is the minimum cardinality of a connected resolving set in a graph G, denoted by cr(G). A cr-set of G is a connected resolving set with cardinality cr(G). A connected graph H is a resolving graph if there is a graph G with a cr-set W such that = H. Let {G_i} be a finite collection of graphs and each G_i has a fixed vertex v_{oi} called a terminal. The amalgamation Amal{Gi,v_{oi}} is formed by taking of all the G_i's and identifying their terminals. In this paper, we determine the connected resolving number and characterize the resolving graphs of amalgamation of cycles.