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A simple linear-time algorithm for generating auxiliary 3-edge-connected subgraphs

Yung H. Tsin

TL;DR

This work addresses computing auxiliary subgraphs for the $3$-edge-connected components ($3ecc$) of a connected multigraph in linear time. It introduces a contraction-like absorb-eject operation executed during a single DFS to merge vertices into $3ecc$s while maintaining invariants that build the corresponding auxiliary subgraphs. The approach extends prior work by producing the $3ecc$ auxiliary subgraphs directly, achieving $O(|V|+|E|)$ time and offering a simpler, single-pass alternative to multi-stage decompositions. This has practical impact as a preprocessing step for higher-connectivity component analysis, including 4-edge-connected decomposition, and broad applicability to graph connectivity studies.

Abstract

A linear-time algorithm for generating auxiliary subgraphs for the 3-edge-connected components of a connected multigraph is presented. The algorithm uses an innovative graph contraction operation and makes only one pass over the graph. By contrast, the previously best-known algorithms make multiple passes over the graph to decompose it into its 2-edge-connected components or 2-vertex-connected components, then its 3-edge-connected components or 3-vertex-connected components, and then construct a cactus representation for the 2-cuts to generate the auxiliary subgraphs for the 3-edge-connected components.

A simple linear-time algorithm for generating auxiliary 3-edge-connected subgraphs

TL;DR

This work addresses computing auxiliary subgraphs for the -edge-connected components () of a connected multigraph in linear time. It introduces a contraction-like absorb-eject operation executed during a single DFS to merge vertices into s while maintaining invariants that build the corresponding auxiliary subgraphs. The approach extends prior work by producing the auxiliary subgraphs directly, achieving time and offering a simpler, single-pass alternative to multi-stage decompositions. This has practical impact as a preprocessing step for higher-connectivity component analysis, including 4-edge-connected decomposition, and broad applicability to graph connectivity studies.

Abstract

A linear-time algorithm for generating auxiliary subgraphs for the 3-edge-connected components of a connected multigraph is presented. The algorithm uses an innovative graph contraction operation and makes only one pass over the graph. By contrast, the previously best-known algorithms make multiple passes over the graph to decompose it into its 2-edge-connected components or 2-vertex-connected components, then its 3-edge-connected components or 3-vertex-connected components, and then construct a cactus representation for the 2-cuts to generate the auxiliary subgraphs for the 3-edge-connected components.
Paper Structure (5 sections, 8 theorems, 3 figures)

This paper contains 5 sections, 8 theorems, 3 figures.

Table of Contents

  1. Introduction
  2. Basic definitions and facts
  3. A linear-time algorithm for generating the auxiliary 3ecc subgraphs
  4. Generating the auxiliary 3ecc subgraphs
  5. Conclusion

Key Result

Theorem 1.1

Let $G=(V,E)$ be a connected graph. For each 3ecc of $G$, replace every 2-cut $\{(v,w),(\ddot{v},\ddot{w})\}$ in $G$, where $v$ and $\ddot{v}$ are vertices in the 3ecc such that $v \neq \ddot{v}$ with a new edge $(v,\ddot{v})$. Let the resulting graph be $\acute{G}$. Then, $\forall x,y \in V, x \ove

Figures (3)

  • Figure 1: A graph and its auxiliary 3ecc subgraphs; dotted lines are auxiliary edges
  • Figure 2: The absorb-eject operation: $(i)$$deg_{G'}(u) = 2$; $(ii)$ edge $(w,u)$ is not a cut-edge.
  • Figure 3: $(i)$ Palm trees: $T_{u_1}^{\curvearrowright}, T_{u_2}^{\curvearrowright}, T_{u_3}^{\curvearrowright}$; $(ii)$ Transformed palm trees: $\dot{T}_{u_1}^{\curvearrowright}, \dot{T}_{u_2}^{\curvearrowright}, \dot{T}_{u_3}^{\curvearrowright}$; $(iii)$ Extended transformed palm trees: $\ddot{T}_{u_1}^{\curvearrowright}, \ddot{T}_{u_2}^{\curvearrowright}, \ddot{T}_{u_3}^{\curvearrowright}$, and transformed palm tree $\dot{T}_w^{\curvearrowright}$. $ear(w) = ear_{\dot{T}_{w}^{\curvearrowright}}(w) = \min_{\lessdot}\{ ear_{\ddot{T}_{\tilde{x}}^{\curvearrowright}}(\tilde{x}), ear_{\ddot{T}_{u_1}^{\curvearrowright}}(w), ear_{\ddot{T}_{u_2}^{\curvearrowright}}(w), ear_{\ddot{T}_{u_3}^{\curvearrowright}}(w)$$= \min_{\lessdot}\{ (w \curvearrowright x),ear_{\ddot{T}_{u_1}^{\curvearrowright}}(w),$$ear_{\ddot{T}_{u_2}^{\curvearrowright}}(w), (w \curvearrowright t(ear(u_3))) \}$$= \min_{\lessdot}\{ (w \curvearrowright x), \hat{f_1}, f_2, (w \curvearrowright z) \} = (w \curvearrowright z).$

Theorems & Definitions (8)

  • Theorem 1.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Theorem 3.6
  • Theorem 3.7