Metric Dimension of Amalgamation of Regular Graphs

الأصل · EN

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 Vertex-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 edge-amalgamation for finite collection of regular graphs: complete graphs and prisms.