On the distance pattern distinguishing number of a graph (Q2336345)

Summary: 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.