Imran Javaid

On the metric dimension of circulant graphs

Abstract Let G = ( V , E ) be a connected graph and d ( x , y ) be the distance between the vertices x and y in V ( G ) . A subset of vertices W = { w 1 , w 2 , … , w k } is called a resolving set or locating set for G if for every two distinct vertices x , y ∈ V ( G ) , there is a vertex w i ∈ W such that d ( x , w i ) ≠ d ( y , w i ) for i = 1 , 2 , … , k . A resolving set containing the minimum number of vertices is called a metric basis for G and the number of vertices in a metric basis is its metric dimension, denoted by d i m ( G ) . Let F be a family of connected graphs G n : F = ( G n ) n ≥ 1 depending on n as follows: the order | V ( G ) | = φ ( n ) and lim n → ∞ φ ( n ) = ∞ . If there exists a constant C > 0 such that d i m ( G n ) ≤ C for every n ≥ 1 then we shall say that F has bounded metric dimension. The metric dimension of a class of circulant graphs C n ( 1 , 2 ) has been determined by Javaid and Rahim (2008) [13] . In this paper, we extend this study to an infinite class of circulant graphs C n ( 1 , 2 , 3 ) . We prove that the circulant graphs C n ( 1 , 2 , 3 ) have metric dimension equal to 4 for n ≡ 2 , 3 , 4 , 5 ( mod 6 ) . For n ≡ 0 ( mod 6 ) only 5 vertices appropriately chosen suffice to resolve all the vertices of C n ( 1 , 2 , 3 ) , thus implying that d i m ( C n ( 1 , 2 , 3 ) ) ≤ 5 except n ≡ 1 ( mod 6 ) when d i m ( C n ( 1 , 2 , 3 ) ) ≤ 6 .

On the metric dimension of generalized Petersen graphs

Abstract A family G of connected graphs is said to be a family with constant metric dimension if its metric dimension is finite and does not depend upon the choice of G in G. In this paper, we study the metric dimension of the generalized Petersen graphs P(2m, m − 1) and give a partial answer to an open problem raised in [13]: Is the generalized Petersen graphs P(s, t), for s ≥ 7 and 3 ≤ t ≤ , a family of graphs with constant metric dimension? We prove that the generalized Petersen graphs P(2m, m − 1) have metric dimension equal to 3 for all odd m ≥ 3, and equal to 4 for all even m ≥ 4.

Locating-dominating sets in hypergraphs

A hypergraph is a generalization of a graph where edges can connect any number of vertices. In this paper, we extend the study of locating-dominating sets to hypergraphs. Along with some basic results, sharp bounds for the location-domination number of hypergraphs in general and exact values with specified conditions are investigated. Moreover, locating-dominating sets in some specific hypergraphs are found.

On the locatic number of graphs

A vertex v of a connected graph G distinguishes a pair u, w of vertices of G if d(v, u)≠d(v, w), where d(·,·) denotes the length of a shortest path between two vertices in G. A k-partition Π={S 1, S 2, …, S k } of the vertex set of G is said to be a locatic partition if for every pair of distinct vertices v and w of G, there exists a vertex s∈S i for all 1≤i≤k that distinguishes v and w. The cardinality of a largest locatic partition is called the locatic number of G. In this paper, we study the locatic number of paths, cycles and characterize all the connected graphs of order n having locatic number n, n−1 and n−2. Some realizable results are also given in this paper.

On the Distinguishing Number of Functigraphs

Let

Bounds on the domination number and the metric dimension of co-normal product of graphs

G_{1}

Resolvability in Circulant Graphs

and

Metric Dimension and Determining Number of Cayley Graphs

G_{2}

On the Metric Dimension of Generalized Petersen Graphs.

be disjoint copies of a graph

On the constant metric dimension of generalized petersen graphs P( n , 4)

G