Metric Dimension of Amalgamation of Regular Graphs

Rinovia Simanjuntak; Danang Tri Murdiansyah

paper

Metric Dimension of Amalgamation of Regular Graphs

arXiv:1401.5164

Abstract

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 ${G_{1}, G_{2}, \dots, G_{n}}$ be a finite collection of graphs and each $G_{i}$ has a fixed vertex $v_{0_{i}}$ or a fixed edge $e_{0_{i}}$ called a terminal vertex or edge, respectively. The vertex-amalgamation of $G_{1}, G_{2}, \dots, G_{n}$ , denoted by $V er t e x - A ma l {G_{i}; v_{0_{i}}}$ , is formed by taking all the $G_{i}$ 's and identifying their terminal vertices. Similarly, the edge-amalgamation of $G_{1}, G_{2}, \dots, G_{n}$ , denoted by $E d g e - A ma l {G_{i}; e_{0_{i}}}$ , is formed by taking all the $G_{i}$ '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.

arXiv comment9 pages