Forcing 2-Metric Dimension in the Join and Corona of Graphs

Abstract

Let G be an undirected and connected graph with vertex set V(G). An ordered set of vertices {x1,...,xk} is a 2-resolving set in G if, for each distinct vertices u,v ∈ V(G), the lists of distances (dG(u,x1),..., dG(u,xk)) and (dG(v,x1),..., dG(v,xk)) differ in at least 2 positions. The minimum size of a 2-resolving set is the 2-metric dimension dim2(G) of G. A 2-resolving set of size dim2(G) is called a 2-metric basis for G. A subset S of a 2-metric basis W of G with the property that W is the unique 2-metric basis containing S is called a forcing subset of W. The forcing number fdim2(W) of W is the minimum cardinality of forcing subsets of W. The forcing number fdim2(G) of G is the smallest forcing number among all 2-metric basis of G.
 This study deals with the forcing subsets of 2-metric basis in graphs. The 2-metric basis in graphs resulting from some binary operations such as join and corona of graphs have been characterized. These characterizations are used to determine values for the forcing number of the 2-metric dimension of each graph considered.