Metric dimension for random graphs

The metric dimension of a graph $G$ is the minimum number of vertices in a subset $S$ of the vertex set of $G$ such that all other vertices are uniquely determined by their distances to the vertices in $S$. In this paper we investigate the metric dimension of the random graph $G(n,p)$ for a wide range of probabilities $p=p(n)$

Metric Dimension of Amalgamation of Graphs

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 \emph{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 \emph{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.Comment: 9 pages, 2 figures, Seventh Czech-Slovak International Symposium on Graph Theory, Combinatorics, Algorithms and Applications (CSGT2013), revised version 21 December 201

Automatic Aircraft Navigation Using Star Metric Dimension Theory in Fire Protected Forest Areas

The purpose of this research is to determine the navigation of an unmanned aircraft automatically using theory of the metric dimension of stars in a forest fire area. The research will also be expanded by determining the star metric dimensions on other unique graphs and graphs resulting from amalgamation operations. The methods used in this research are pattern recognition and axiomatic deductive methods. The pattern detection method is to look for patterns to construct differentiated sets on the metric dimension (dim) so that the coordinate values are minimum and different. Meanwhile, axiomatic deductive is a research method that uses deductive proof principles that apply in mathematical logic by using existing axioms or theorems to solve a problem. Then the method is used to determine the stars' metric dimensions

A Simple Algorithm for Graph Reconstruction

How efficiently can we find an unknown graph using distance queries between its vertices? We assume that the unknown graph is connected, unweighted, and has bounded degree. The goal is to find every edge in the graph. This problem admits a reconstruction algorithm based on multi-phase Voronoi-cell decomposition and using $\tilde O(n^{3/2})$ distance queries. In our work, we analyze a simple reconstruction algorithm. We show that, on random $\Delta$-regular graphs, our algorithm uses $\tilde O(n)$ distance queries. As by-products, we can reconstruct those graphs using $O(\log^2 n)$ queries to an all-distances oracle or $\tilde O(n)$ queries to a betweenness oracle, and we bound the metric dimension of those graphs by $\log^2 n$. Our reconstruction algorithm has a very simple structure, and is highly parallelizable. On general graphs of bounded degree, our reconstruction algorithm has subquadratic query complexity

On the limiting distribution of the metric dimension for random forests

The metric dimension of a graph G is the minimum size of a subset S of vertices of G such that all other vertices are uniquely determined by their distances to the vertices in S. In this paper we investigate the metric dimension for two different models of random forests, in each case obtaining normal limit distributions for this parameter.Comment: 22 pages, 5 figure