On Minimum Metric Dimension of Circulant Networks
نویسنده
چکیده
Let M = } ,..., , { 2 1 n v v v be an ordered set of vertices in a graph G. Then )) , ( ),..., , ( ), , ( ( 2 1 n v u d v u d v u d is called the M-coordinates of a vertex u of G. The set M is called a metric basis if the vertices of G have distinct M-coordinates. A minimum metric basis is a set M with minimum cardinality. The cardinality of a minimum metric basis of G is called minimum metric dimension and is denoted by (G). This concept has wide applications in motion planning and in the field of robotics. In this paper we determine the minimum metric dimension of certain classes of circulant networks. We prove that for circulant graphs G(n; {1, 2}), (G (n; {1, 2}) = 3, when n = 4 l, 4 l + 2, 4 l + 3, l 1 and 2 < (G (n; ±{1, 2}) 4, for 4 l+ 1, l 1. We have similar results for circulant digraphs G(n; {1, 2, 3}) and certain subclasses of circulant graphs.
منابع مشابه
The metric dimension of the circulant graph C(n, \pm\{1, 2, 3, 4\})
Let G = (V, E) be a connected graph and let d(u, v) denote the distance between vertices u, v ∈ V. A metric basis for G is a set B ⊆ V of minimum cardinality such that no two vertices of G have the same distances to all points of B. The cardinality of a metric basis of G is called the metric dimension of G, denoted by dim(G). In this paper we determine the metric dimension of the circulant grap...
متن کاملThe metric dimension of circulant graphs and Cayley hypergraphs
Let G = (V,E) be a connected graph (or hypergraph) and let d(x, y) denote the distance between vertices x, y ∈ V (G). A subset W ⊆ V (G) is called a resolving set for G if for every pair of distinct vertices x, y ∈ V (G), there is w ∈ W such that d(x,w) 6= d(y, w). The minimum cardinality of a resolving set for G is called the metric dimension of G, denoted by β(G). In this paper we determine t...
متن کاملThe Metric Dimension of Circulant Graphs and Their Cartesian Products
Let G = (V,E) be a connected graph (or hypergraph) and let d(x, y) denote the distance between vertices x, y ∈ V (G). A subset W ⊆ V (G) is called a resolving set for G if for every pair of distinct vertices x, y ∈ V (G), there is w ∈ W such that d(x,w) 6= d(y, w). The minimum cardinality of a resolving set for G is called the metric dimension of G, denoted by β(G). The circulant graph Cn(1, 2,...
متن کاملOn two-dimensional Cayley graphs
A subset W of the vertices of a graph G is a resolving set for G when for each pair of distinct vertices u,v in V (G) there exists w in W such that d(u,w)≠d(v,w). The cardinality of a minimum resolving set for G is the metric dimension of G. This concept has applications in many diverse areas including network discovery, robot navigation, image processing, combinatorial search and optimization....
متن کاملSelected Topics in the Extremal Graph Theory
Extremal problems in graph theory form a very wide research area. We study the following topics: the metric dimension of circulant graphs, the Wiener index of trees of given diameter, and the degree‐diameter problem for Cayley graphs. All three topics are connected to the study of distances in graphs. We give a short survey on the topics and present several new results.
متن کامل