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

Yiyang Jiang, Xudong Chen

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.

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

Abstract

In this paper, we focus on the class of complete -partite graphs, for an undirected graph possibly with self-loops, and address the problem of finding largest -regular subgraphs of these graphs, which can be formulated as an integer linear program. Roughly speaking, a complete -partite graph is obtained by replacing every single node of with a number of nodes, preserving the edge/non-edge relations of . Our motivation in finding largest -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 , independent of the order/size of the -partite graph itself. Furthermore, we demonstrate through simulations that with high probability, a random -partite graph contains a largest -regular subgraph of the same order as its complete counterpart does.

Paper Structure

This paper contains 13 sections, 10 theorems, 34 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivation and relevance of our problem
  3. Literature review
  4. Notation
  5. Problem Formulation and Main Result
  6. Proof of Items 1 and 2 of Theorem \ref{['thm:main']}
  7. Half-integrality of the Linear Program
  8. Proof of item 1 of Theorem \ref{['thm:main']}
  9. Proof of Item 2 of Theorem \ref{['thm:main']}
  10. Hamiltonicity of Complete $S$-partite Graph
  11. Proof of Item 3 of Theorem \ref{['thm:main']}
  12. Numerical Study
  13. Conclusions

Key Result

Theorem 1

Let $x\in \mathbb{N}^q$ and $S$ be an undirected graph on $q$ nodes, possibly with self-loops. Then, the following hold:

Figures (7)

  • Figure 1: (a) Skeleton graph $S$. (b) The associated bipartite graph $B$. The highlighted edges show the correspondence between $S$ and $B$. We also present $Z$ and $\hat{Z}$ the incidence matrices.
  • Figure 2: (a) The auxiliary pseudograph $M$ for Example \ref{['exmp:1']}, with blue edges labeled by their order in $\mathcal{E}$. (b) The corresponding complete $S$-partite graph $K_y$, with the lifted Hamilton cycle $H$ shown in blue.
  • Figure 3: An instance in which the support graph $S_c$ has two nontrivial connected components. (a) The skeleton $S$. (b) The support graph $S_c$. (c) The associated pseudograph $M_c$. (d) The blow-up $K_y$ and the resulting cycle cover $H\subseteq K_y$.
  • Figure 4: The skeleton graph $S$ considered in the numerical study.
  • Figure 5: Empirical PMFs $\widehat{\mathbb{P}}(n(G)=t)$ for the setting $x := (24,7,4,11,6,4)$ with $n = 56$. The dashed, vertical line is at $n^* = 42$. Values of $\hat{p}^*$ for different $p$ are reported.
  • ...and 2 more figures

Theorems & Definitions (21)

  • Definition 1: $S$-partite graph
  • Theorem 1
  • Proposition 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • Proposition 3
  • Lemma 3
  • ...and 11 more