Computing the number of induced copies of a fixed graph in a bounded degree graph

TL;DR

This paper presents an algorithm to efficiently count the number of induced subgraphs isomorphic to a fixed graph in bounded degree graphs, with a runtime that is nearly optimal and depends exponentially on the size of the fixed graph.

Contribution

The authors develop a nearly optimal algorithm for counting induced copies of a fixed graph in bounded degree graphs with a runtime exponential in the size of the fixed graph.

Findings

Algorithm runs in O(c^m * n) time for fixed graph of size m

Runtime is essentially optimal given the problem's complexity

Applicable to graphs with bounded maximum degree

Abstract

In this paper we show that for any graph $H$ of order $m$ and any graph $G$ of order $n$ and maximum degree $\Delta$ one can compute the number of subsets $S$ of $V(G)$ that induces a graph isomorphic to $H $in time $O(c^m\cdot n)$ for some constant $c = c(\Delta) > 0$. This is essentially best possible.