Efficient Enumeration of Bipartite Subgraphs in Graphs

TL;DR

This paper introduces the first efficient algorithms for enumerating all bipartite subgraphs in a graph, significantly advancing the computational methods for this fundamental graph class.

Contribution

It presents two novel algorithms for bipartite subgraph enumeration, one with $O(k)$ time per solution and another with $O(1)$ time per solution.

Findings

Algorithms enumerate all bipartite induced subgraphs efficiently

Enumeration time is $O(k)$ per solution for induced subgraphs

Enumeration time is $O(1)$ per solution for all bipartite subgraphs

Abstract

Subgraph enumeration problems ask to output all subgraphs of an input graph that belongs to the specified graph class or satisfy the given constraint. These problems have been widely studied in theoretical computer science. As far, many efficient enumeration algorithms for the fundamental substructures such as spanning trees, cycles, and paths, have been developed. This paper addresses the enumeration problem of bipartite subgraphs. Even though bipartite graphs are quite fundamental and have numerous applications in both theory and application, its enumeration algorithms have not been intensively studied, to the best of our knowledge. We propose the first non-trivial algorithms for enumerating all bipartite subgraphs in a given graph. As the main results, we develop two efficient algorithms: the one enumerates all bipartite induced subgraphs of a graph with degeneracy $k$ in $O(k)$ time per solution. The other enumerates all bipartite subgraphs in $O(1)$ time per solution.