On the Distance Pattern Distinguishing Number of a Graph

Abstract

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.