Metric Dimension for Amalgamations of Graphs
نویسندگان
چکیده
A set of vertices S resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set of G. Let {G1, G2, . . . , Gn} be a finite collection of graphs and each Gi has a fixed vertex v0i or a fixed edge e0i called a terminal vertex or edge, respectively. The vertex-amalgamation of G1, G2, . . . , Gn, denoted by V ertex−Amal{Gi; v0i}, is formed by taking all the Gi’s and identifying their terminal vertices. Similarly, the edge-amalgamation of G1, G2, . . . , Gn, denoted by Edge−Amal{Gi; e0i}, is formed by taking all the Gi’s and identifying their terminal edges. Here we study the metric dimensions of vertex-amalgamation and edgeamalgamation for finite collection of arbitrary graphs. We give lower and upper bounds for the dimensions, show that the bounds are tight, and construct infinitely many graphs for each possible value between the bounds.
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