Counting Subgraphs in Degenerate Graphs

TL;DR

This paper characterizes exactly which fixed graphs can be counted efficiently in bounded degeneracy graphs, resolving a long-standing open problem and extending understanding of subgraph counting complexity.

Contribution

It proves that a previously known sufficient condition for easy subgraph counting is also necessary, fully characterizing easy graphs for counting copies, induced copies, and homomorphisms.

Findings

Complete characterization of easy graphs for counting subgraphs.

Extension of results to counting induced subgraphs and homomorphisms.

Resolution of the open problem posed by Bera, Pashanasangi, and Seshadhri.

Abstract

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 $k > 5$ 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.