Abstract
The status of a vertex v in a connected graph is the sum of the distances from v to all other vertices. The status sequence of a connected graph is the list of the statuses of all the vertices of the graph. In this paper we investigate the status sequences of trees. Particularly, we show that it is NP-complete to decide whether there exists a tree that has a given sequence of integers as its status sequence. We also present some new results about trees whose status sequences are comprised of a few distinct numbers or many distinct numbers. In this direction, we show that any status injective tree is unique among trees. Finally, we investigate how orbit partitions and equitable partitions relate to the status sequence.
Highlights
Sequences associated with a graph, such as the degree sequence, spectrum, and status sequence, contain useful information about the graph’s structure and give a compact representation of the graph without using vertex adjacencies
Status sequences have been used in interesting practical applications, such as efficiently determining whether a company sells a particular type of circuit board; in [2], it was shown that algorithms for board equivalence based on graph invariants such as the status sequence require much less time and memory compared to algorithms that use the complete topology of the circuit boards
We find that there is a unique tree with the given status sequence in this case, if one exists
Summary
Sequences associated with a graph, such as the degree sequence, spectrum, and status sequence, contain useful information about the graph’s structure and give a compact representation of the graph without using vertex adjacencies. We study the status sequences of trees, and answer several questions about status realizability, uniqueness, and their relation to various graph partitions. Sequences related to distances in a graph have been studied, see, e.g., [8] In those papers, the authors tackle the recognition problems, and construct fast algorithms for finding graphs that realize given sequences. We show that in the highly asymmetric cases with k(G) = n, if a tree realizing a given status sequence exists, this tree is uniquely defined. In this sense, we explore the complexity landscape of the status sequence problem in trees based on tree symmetries.
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