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Low-degree spanning trees of $2$-edge-connected graphs in linear time

Dariusz Dereniowski, Janusz Dybizbański, Przemysław Karpiński, Michał Zakrzewski, Paweł Żyliński

Abstract

We present a simple linear-time algorithm that finds a spanning tree $T$ of a given $2$-edge-connected graph $G$ such that each vertex $v$ of $T$ has degree at most $\lceil \frac{°_G(v)}{2}\rceil + 1$.

Low-degree spanning trees of $2$-edge-connected graphs in linear time

Abstract

We present a simple linear-time algorithm that finds a spanning tree of a given -edge-connected graph such that each vertex of has degree at most .

Paper Structure

This paper contains 7 sections, 6 theorems, 8 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Problem statement and our result
  2. Paper organization.
  3. Background.
  4. Preliminaries
  5. The algorithm
  6. The complexity and correctness
  7. Vertex degrees in $T$

Key Result

Theorem 1

There exists a $\mathcal{O}(m)$-time algorithm that for any $2$-edge-connected graph $G=(V_G,E_G)$ finds its spanning tree $T$ such that for each vertex $v \in V_G$, it holds

Figures (1)

  • Figure : (input: a graph $G$; output: a spanning tree $T$ of $G$)

Theorems & Definitions (9)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof