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A note on the number of non-cycle components in a pseudo 2-factor of graphs

Masaki Kashima

TL;DR

The paper studies how many non-cycle components can appear in a pseudo 2-factor of a graph. It introduces $f(G) = \max\{|I| - \delta_G(I) + 1 : I \text{ independent}\}$ and proves that every graph $G$ has a pseudo 2-factor with at most $\max\{0, f(G)\}$ non-cycle components, unifying and extending prior results related to the number of non-cycle components. The proof combines a forest-based bound with a careful augmentation argument on a maximal union of disjoint cycles to show $\alpha(H) \le f(G)$ for the remaining forest $H$, or else produce a larger 2-regular subgraph, contradicting maximality. The work also discusses algorithmic aspects, giving a polynomial-time method to obtain the bound but highlighting limitations in achieving the minimum possible non-cycle components and leaving an open question about a polynomial-time algorithm for that optimization task.

Abstract

A pseudo 2-factor of a graph is a spanning subgraph such that each component is $K_1$, $K_2$, or a cycle. This notion was introduced by Bekkai and Kouider in 2009, where they showed that every graph $G$ has a pseudo 2-factor with at most $α(G)-δ(G)+1$ components that are not cycles. For a graph $G$ and a set of vertices $S$, let $δ_G(S)$ denote the minimum degree of vertices in $S$. In this note, we show that every graph $G$ has a pseudo 2-factor with at most $f(G)$ components that are not cycles, where $f(G)$ is the maximum value of $|I|-δ_G(I)+1$ among all independent sets $I$ of $G$. This result is a common generalization of a result by Bekkai and Kouider and a previous result by the author on the existence of a 2-factor.

A note on the number of non-cycle components in a pseudo 2-factor of graphs

TL;DR

The paper studies how many non-cycle components can appear in a pseudo 2-factor of a graph. It introduces and proves that every graph has a pseudo 2-factor with at most non-cycle components, unifying and extending prior results related to the number of non-cycle components. The proof combines a forest-based bound with a careful augmentation argument on a maximal union of disjoint cycles to show for the remaining forest , or else produce a larger 2-regular subgraph, contradicting maximality. The work also discusses algorithmic aspects, giving a polynomial-time method to obtain the bound but highlighting limitations in achieving the minimum possible non-cycle components and leaving an open question about a polynomial-time algorithm for that optimization task.

Abstract

A pseudo 2-factor of a graph is a spanning subgraph such that each component is , , or a cycle. This notion was introduced by Bekkai and Kouider in 2009, where they showed that every graph has a pseudo 2-factor with at most components that are not cycles. For a graph and a set of vertices , let denote the minimum degree of vertices in . In this note, we show that every graph has a pseudo 2-factor with at most components that are not cycles, where is the maximum value of among all independent sets of . This result is a common generalization of a result by Bekkai and Kouider and a previous result by the author on the existence of a 2-factor.
Paper Structure (4 sections, 6 theorems, 9 equations, 3 figures)

This paper contains 4 sections, 6 theorems, 9 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Proofs
  3. Proof of Theorem \ref{['thm:main']}
  4. Remarks on algorithmic aspect of pseudo 2-factor

Key Result

Theorem 1

For any graph $G$ with $\alpha(G)\geq \delta(G)$, $G$ has a pseudo 2-factor with at most $\alpha(G)-\delta(G)+1$ non-cycle components.

Figures (3)

  • Figure 1: A graph $G_2$ which has a gap between $f(G_2)$ and $\alpha(G_2)-\delta(G_2)+1$.
  • Figure 2: A pseudo 2-factor with minimum number of non-cycle components.
  • Figure 3: A maximum 2-regular subgraph.

Theorems & Definitions (10)

  • Theorem 1: BK2009
  • Theorem 2: N1995
  • Theorem 3: Karxiv
  • Theorem 4
  • Proposition 6
  • proof
  • Proposition 7
  • proof
  • proof
  • proof