A Linear Delay Algorithm for Enumeration of 2-Edge/Vertex-connected Induced Subgraphs

TL;DR

This paper introduces a linear delay algorithm for efficiently enumerating all 2-edge and 2-vertex-connected induced subgraphs, advancing the computational methods for analyzing graph connectivity structures.

Contribution

It presents the first linear delay enumeration algorithms for all 2-edge-connected and 2-vertex-connected induced subgraphs using a novel partition-based approach.

Findings

First linear delay algorithms for enumerating 2-edge-connected induced subgraphs.

First linear delay algorithms for enumerating 2-vertex-connected induced subgraphs.

Efficient enumeration method based on a new set system property.

Abstract

For a set system $(V,{\mathcal C}\subseteq 2^V)$, we call a subset $C\in{\mathcal C}$ a component. A nonempty subset $Y\subseteq C$ is a minimal removable set (MRS) of $C$ if $C\setminus Y\in{\mathcal C}$ and no proper nonempty subset $Z\subsetneq Y$ satisfies $C\setminus Z\in{\mathcal C}$. In this paper, we consider the problem of enumerating all components in a set system such that, for every two components $C,C'\in{\mathcal C}$ with $C'\subsetneq C$, every MRS $X$ of $C$ satisfies either $X\subseteq C'$ or $X\cap C'=\emptyset$. We provide a partition-based algorithm for this problem, which yields the first linear delay algorithms to enumerate all 2-edge-connected induced subgraphs, and to enumerate all 2-vertex-connected induced subgraphs.