This paper is devoted to the study of the arithmetic graph of a composite number m, denoted by A m . It has been observed that there exist different composite numbers for which the arithmetic graphs are isomorphic. It is proved that the maximum distance between any two vertices of A m is two or three. Conditions under which the vertices have the sa...
In this paper we focus on the issue related to finding the resolving connected dominating sets (RCDSs) of a graph, denoted by G. The connected dominating set (CDS) is a connected subset of vertices of G selected to guarantee that all vertices in the graph are connected to vertices in the CDS. The connected dominating set with minimum cardinality, o...
Let G = ( V ( G ) , E ( G ) ) be a connected graph. An ordered set W &sub / V ( G ) is a resolving set for G if every vertex of G is uniquely determined by its vector of distances to the vertices in W. The metric dimension of G is the minimum cardinality of a resolving set. In this paper, we characterize the graphs of metric dimension n &minus / 3 ...
In this paper, we consider fault-tolerant resolving sets in graphs. We characterize n-vertex graphs with fault-tolerant metric dimension n, n &minus / 1 , and 2, which are the lower and upper extremal cases. Furthermore, in the first part of the paper, a method is presented to locate fault-tolerant resolving sets by using classical resolving sets i...
Let G = (V, E) be a connected graph, let x ∈ V (G) be a vertex and e = yz ∈ E(G) be an edge. The distance between the vertex x and the edge e is given by d G (x, e) = min{d G (x, y), d G (x, z)}. A vertex t ∈ V (G) distinguishes two edges e, f ∈ E(G) if d G (t, e) = d G (t, f). A set R ⊆ V (G) is an edge metric generator for G if every two edges of...
Let G = (V, E) be a connected graph and d(x, y) be the distance between the vertices x and y in G. A set of vertices W resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in W. A metric dimension of G is the minimum cardinality of a resolving set of G and is denoted by dim(G). In this paper, Cycle, P...
Let G = (V, E) be a simple, finite, and connected graph. A subset S = {u1, u2, &hellip / , uk} of V(G) is called a resolving set (locating set) if for any x &isin / V(G), the code of x with respect to S that is denoted by CS (x), which is defined as CS (x) = (d(u1, x), d(u2, x), .., d(uk, x)), is different for different x. The minimum cardinality o...
Concepts of resolving set and metric basis has enjoyed a lot of success because of multi-purpose applications both in computer and mathematical sciences. For a connected graph G(V,E) a subset W of V(G) is a resolving set for G if every two vertices of G have distinct representations with respect to W. A resolving set of minimum cardinality is calle...
Let G = (V, E) be a connected graph and d(u, v) denote the distance between the vertices u and v in G. A set of vertices W resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in W. A metric dimension of G is the minimum cardinality of a resolving set of G and is denoted by dim(G). Let J2n,m be a m-le...
Recently upper bounds on the metric dimension of Grassmann graphs, bilinear forms graphs, doubled Grassmann graphs and twisted Grassmann graphs have been obtained by Bailey, Meagher, Feng and the present authors. In this paper, we continue this research and construct resolving sets with smaller sizes for these graphs. As a result, we improve the kn...
