On the Distance Pattern Distinguishing Number of a Graph

Let G=(V,E) be a connected simple graph and let M be a nonempty subset of V. The M-distance pattern of a vertex u in G is the set of all distances from u to the vertices in M. If the distance patterns of all vertices in V are distinct, then the set M is a distance pattern distinguishing set of G. A graph G with a distance pattern distinguishing set is called a distance pattern distinguishing graph. Minimum number of vertices in a distance pattern distinguishing set is called distance pattern distinguishing number of a graph. This paper initiates a study on the problem of finding distance pattern distinguishing number of a graph and gives bounds for distance pattern distinguishing number. Further, this paper provides an algorithm to determine whether a graph is a distance pattern distinguishing graph or not and hence to determine the distance pattern distinguishing number of that graph