Forcibly bipartite and acyclic (uni-)graphic sequences

Amotz Bar-Noy, Toni Böhnlein, David Peleg, Dror Rawitz

Research output: Contribution to journalArticlepeer-review

Abstract

The paper concerns the question of which properties of a graph are already determined by its degree sequence. The classic degree realization problem asks to characterize graphic sequences, i.e., sequences of positive integers, which are the degree sequence of some simple graph. Erdős and Gallai [11] solved this problem. Havel and Hakimi [14,17] provide a different characterization implying an algorithm to generate a realizing graph (if one exists). It is known that a graphic sequence can have several non-isomorphic realizations. We characterize graphic sequences where every realization has some given graph property P. Such sequences are called forcibly P-graphic. In particular, we present complete results characterizing forcibly (connected) bipartite, forcibly acyclic, and forcibly tree-graphic sequences. In those four models, we also characterize the sequences with a unique realizing graph, called unigraphic sequences. Finally, we address the problem of counting the number of sequences in each model.
Original languageEnglish
Article number113460
Number of pages8
JournalDiscrete Mathematics
Volume346
Issue number7
DOIs
StatePublished - Jul 2023

Keywords

  • Acyclic graphs
  • Bipartite graphs
  • Degree sequence
  • Forcibly P-graphic
  • Graph realization
  • Unique P-graphic

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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