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

FAULT-TOLERANT METRIC DIMENSION OF CIRCULANT GRAPHS

A set $W$ of vertices in a graph $G$ is called a resolving setfor $G$ if for every pair of distinct vertices $u$ and $v$ of $G$ there exists a vertex $w \in W$ such that the distance between $u$ and $w$ is different from the distance between $v$ and $w$. The cardinality of a minimum resolving set is called the metric dimension of $G$, denoted by $\beta(G)$. A resolving set $W'$ for $G$ is fault-tolerant if $W'\setminus \left\lbrace w\right\rbrace$ for each $w$ in $W'$, is also a resolving set and the fault-tolerant metric dimension of $G$ is the minimum cardinality of such a set, denoted by $\beta'(G)$. The circulant graph is a graph with vertex set $\mathbb{Z}_{n}$, an additive group of integers modulo $n$, and two vertices labeled $i$ and $j$ adjacent if and only if $i -j \left( mod \ n \right)  \in C$, where $C \in \mathbb{Z}_{n}$ has the property that $C=-C$ and $0 \notin C$. The circulant graph is denoted by $X_{n,\bigtriangleup}$ where $\bigtriangleup = \vert C\vert$. In this paper, we study the fault-tolerant metric dimension of a family of circulant graphs $X_{n,3}$ with connection set $C=\lbrace 1,\dfrac{n}{2},n-1\rbrace$ and circulant graphs $X_{n,4}$ with connection set $C=\lbrace \pm 1,\pm 2\rbrace$

Further new results on strong resolving partitions for graphs

A set W of vertices of a connected graph G strongly resolves two different vertices x, y is not an element of W if either d(G) (x, W) = d(G) (x, y) + d(G) (y, W) or d(G) (y, W) = d(G )(y, x) + d(G) (x, W), where d(G) (x, W) = min{d(x,w): w is an element of W} and d (x,w) represents the length of a shortest x - w path. An ordered vertex partition Pi = {U-1, U-2,...,U-k} of a graph G is a strong resolving partition for G, if every two different vertices of G belonging to the same set of the partition are strongly resolved by some other set of Pi. The minimum cardinality of any strong resolving partition for G is the strong partition dimension of G. In this article, we obtain several bounds and closed formulae for the strong partition dimension of some families of graphs and give some realization results relating the strong partition dimension, the strong metric dimension and the order of graphs