## On the distance pattern distinguishing number of a graph.(English)Zbl 1442.05051

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.

### MSC:

 05C12 Distance in graphs
Full Text:

### References:

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