International Workshop on Combinatorial Algorithms

IWOCA 2014: Combinatorial Algorithms pp 330-337

# Metric Dimension for Amalgamations of Graphs

• Rinovia Simanjuntak
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8986)

## 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 $$\{G_1, G_2, \ldots , G_n\}$$ 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 $$G_1, G_2, \ldots , G_n$$, denoted by $$Vertex-Amal\{G_i;v_{0_i}\}$$, is formed by taking all the $$G_i$$’s and identifying their terminal vertices. Similarly, the edge-amalgamation of $$G_1, G_2, \ldots , G_n$$, denoted by $$Edge-Amal\{G_i;e_{0_i}\}$$, 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 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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© Springer International Publishing Switzerland 2015

## Authors and Affiliations

• Rinovia Simanjuntak
• 1
Email author