Counting Subgraphs in Degenerate Graphs

Suman K. Bera, Lior Gishboliner, Yevgeny Levanzov, C. Seshadhri, Asaf Shapira

Research output: Contribution to journalArticlepeer-review


We consider the problem of counting the number of copies of a fixed graph H within an input graph G. This is one of the most well-studied algorithmic graph problems, with many theoretical and practical applications. We focus on solving this problem when the input G has bounded degeneracy. This is a rich family of graphs, containing all graphs without a fixed minor (e.g., planar graphs), as well as graphs generated by various random processes (e.g., preferential attachment graphs). We say that H is easy if there is a linear-time algorithm for counting the number of copies of H in an input G of bounded degeneracy. A seminal result of Chiba and Nishizeki from '85 states that every H on at most 4 vertices is easy. Bera, Pashanasangi, and Seshadhri recently extended this to all H on 5 vertices and further proved that for every there is a k-vertex H which is not easy. They left open the natural problem of characterizing all easy graphs H.Bressan has recently introduced a framework for counting subgraphs in degenerate graphs, from which one can extract a sufficient condition for a graph H to be easy. Here, we show that this sufficient condition is also necessary, thus fully answering the Bera-Pashanasangi-Seshadhri problem. We further resolve two closely related problems; namely characterizing the graphs that are easy with respect to counting induced copies, and with respect to counting homomorphisms.

Original languageEnglish
Article number23
JournalJournal of the ACM
Issue number3
StatePublished - Jun 2022


  • Graph algorithms
  • degeneracy
  • linear time algorithms
  • subgraph counting


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