Largest $2$-regular Subgraphs in complete $S$-partite Graphs

TL;DR

This paper studies the problem of finding the largest 2-regular subgraphs in complete S-partite graphs, providing an efficient solution method and demonstrating high probability of optimal subgraphs in random instances.

Contribution

The paper introduces an efficient $O(|V(S)|^3)$ algorithm for finding largest 2-regular subgraphs in complete S-partite graphs, independent of the graph size.

Findings

The integer linear program can be solved efficiently in $O(|V(S)|^3)$ time.

Random S-partite graphs likely contain maximum 2-regular subgraphs of the same size as complete graphs.

The approach is motivated by structural systems theory applications.

Abstract

In this paper, we focus on the class of complete $S$-partite graphs, for $S$ an undirected graph possibly with self-loops, and address the problem of finding largest $2$-regular subgraphs of these graphs, which can be formulated as an integer linear program. Roughly speaking, a complete $S$-partite graph is obtained by replacing every single node of $S$ with a number of nodes, preserving the edge/non-edge relations of $S$. Our motivation in finding largest $2$-regular subgraphs is rooted in the structural systems theory, particularly in the problem of finding largest subnetworks that can sustain controllability or asymptotic stability of the corresponding subsystems. A main contribution of the paper is to show that the integer linear problem can be solved efficiently in $O(|V(S)|^3)$, independent of the order/size of the $S$-partite graph itself. Furthermore, we demonstrate through simulations that with high probability, a random $S$-partite graph contains a largest $2$-regular subgraph of the same order as its complete counterpart does.