On the status sequences of trees

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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. (C) 2020 The Author(s). Published by Elsevier B.V.

Original languageEnglish
Pages (from-to)110-120
Number of pages11
JournalTheoretical Computer Science
Volume856
DOIs
Publication statusPublished - 8 Feb 2021

JEL classifications

  • c00 - Mathematical and Quantitative Methods: General

Keywords

  • Complexity
  • Graph algorithms
  • Graph analysis
  • Status sequence
  • Trees
  • DISTANCE
  • GRAPHS
  • ROBBER
  • Graph partition
  • STATUS UNIQUE
  • DIMENSION
  • Tree
  • SPECTRA
  • Status injective

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