Efficient Enumeration of Subgraphs and Induced Subgraphs with Bounded Girth

TL;DR

This paper introduces efficient algorithms for enumerating connected subgraphs with bounded girth, enabling the extraction of sparse structures and addressing related graph problems.

Contribution

It presents two novel algorithms for enumerating vertex and edge induced subgraphs with bounded girth, running in linear amortized time and adaptable to weighted graphs.

Findings

Algorithms run in O(n) amortized time per solution.

Space complexity is O(n^3).

Algorithms can be adapted for weighted graphs and relaxed connectivity.

Abstract

The girth of a graph is the length of its shortest cycle. Due to its relevance in graph theory, network analysis and practical fields such as distributed computing, girth-related problems have been object of attention in both past and recent literature. In this paper, we consider the problem of listing connected subgraphs with bounded girth. As a large girth is index of sparsity, this allows to extract sparse structures from the input graph. We propose two algorithms, for enumerating respectively vertex induced subgraphs and edge induced subgraphs with bounded girth, both running in $O(n)$ amortized time per solution and using $O(n^3)$ space. Furthermore, the algorithms can be easily adapted to relax the connectivity requirement and to deal with weighted graphs. As a byproduct, the second algorithm can be used to answer the well known question of finding the densest $n$-vertex graph(s) of girth $k$.