92 research outputs found

    Metric Dimension of Amalgamation of Graphs

    Full text link
    A set of vertices SS resolves a graph GG if every vertex is uniquely determined by its vector of distances to the vertices in SS. The metric dimension of GG is the minimum cardinality of a resolving set of GG. Let {G1,G2,,Gn}\{G_1, G_2, \ldots, G_n\} be a finite collection of graphs and each GiG_i has a fixed vertex v0iv_{0_i} or a fixed edge e0ie_{0_i} called a terminal vertex or edge, respectively. The \emph{vertex-amalgamation} of G1,G2,,GnG_1, G_2, \ldots, G_n, denoted by VertexAmal{Gi;v0i}Vertex-Amal\{G_i;v_{0_i}\}, is formed by taking all the GiG_i's and identifying their terminal vertices. Similarly, the \emph{edge-amalgamation} of G1,G2,,GnG_1, G_2, \ldots, G_n, denoted by EdgeAmal{Gi;e0i}Edge-Amal\{G_i;e_{0_i}\}, is formed by taking all the GiG_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

    Get PDF
    A set WW of vertices in a graph GG is called a resolving setfor GG if for every pair of distinct vertices uu and vv of GG there exists a vertex wWw \in W such that the distance between uu and ww is different from the distance between vv and ww. The cardinality of a minimum resolving set is called the metric dimension of GG, denoted by β(G)\beta(G). A resolving set WW' for GG is fault-tolerant if W{w}W'\setminus \left\lbrace w\right\rbrace for each ww in WW', is also a resolving set and the fault-tolerant metric dimension of GG is the minimum cardinality of such a set, denoted by β(G)\beta'(G). The circulant graph is a graph with vertex set Zn\mathbb{Z}_{n}, an additive group of integers modulo nn, and two vertices labeled ii and jj adjacent if and only if ij(mod n) Ci -j \left( mod \ n \right)  \in C, where CZnC \in \mathbb{Z}_{n} has the property that C=CC=-C and 0C0 \notin C. The circulant graph is denoted by Xn,X_{n,\bigtriangleup} where =C\bigtriangleup = \vert C\vert. In this paper, we study the fault-tolerant metric dimension of a family of circulant graphs Xn,3X_{n,3} with connection set C={1,n2,n1}C=\lbrace 1,\dfrac{n}{2},n-1\rbrace and circulant graphs Xn,4X_{n,4} with connection set C={±1,±2}C=\lbrace \pm 1,\pm 2\rbrace

    Further new results on strong resolving partitions for graphs

    Get PDF
    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
    corecore