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Upper bounds on minimum size of feedback arc set of directed multigraphs with bounded degree

Gregory Gutin, Hui Lei, Anders Yeo, Yacong Zhou

Abstract

An oriented multigraph is a directed multigraph without directed 2-cycles. Let ${\rm fas}(D)$ denote the minimum size of a feedback arc set in an oriented multigraph $D$. The degree of a vertex is the sum of its out- and in-degrees. In several papers, upper bounds for ${\rm fas}(D)$ were obtained for oriented multigraphs $D$ with maximum degree upper-bounded by a constant. Hanauer (2017) conjectured that ${\rm fas}(D)\le 2.5n/3$ for every oriented multigraph $D$ with $n$ vertices and maximum degree at most 5. We prove a strengthening of the conjecture: ${\rm fas}(D)\le m/3$ holds for every oriented multigraph $D$ with $m$ arcs and maximum degree at most 5. This bound is tight and improves a bound of Berger and Shor (1990,1997). It would be interesting to determine $c$ such that ${\rm fas}(D)\le cn$ for every oriented multigraph $D$ with $n$ vertices and maximum degree at most 5 such that the bound is tight. We show that $\frac{5}{7}\le c \le \frac{24}{29} < \frac{2.5}{3}$.

Upper bounds on minimum size of feedback arc set of directed multigraphs with bounded degree

Abstract

An oriented multigraph is a directed multigraph without directed 2-cycles. Let denote the minimum size of a feedback arc set in an oriented multigraph . The degree of a vertex is the sum of its out- and in-degrees. In several papers, upper bounds for were obtained for oriented multigraphs with maximum degree upper-bounded by a constant. Hanauer (2017) conjectured that for every oriented multigraph with vertices and maximum degree at most 5. We prove a strengthening of the conjecture: holds for every oriented multigraph with arcs and maximum degree at most 5. This bound is tight and improves a bound of Berger and Shor (1990,1997). It would be interesting to determine such that for every oriented multigraph with vertices and maximum degree at most 5 such that the bound is tight. We show that .
Paper Structure (6 sections, 10 theorems, 15 equations, 2 figures, 1 table)

This paper contains 6 sections, 10 theorems, 15 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Coefficients in Upper Bounds
  3. Additional Notation and Terminology
  4. Proof of Theorem \ref{['thm:main']}
  5. Proof of Theorem \ref{['thm:5-rg']}
  6. Lower bounds for $c"_{\leq 5}$, $c'_{\leq 6}$ and $c"_{\leq 6}$

Key Result

Theorem 1

Hanauer2017 (i) If $D$ is an oriented multigraph with $\Delta\leq 3$, then ${\rm fas}(D)\leq n/3$. (ii) If $D$ is an oriented multigraph with $\Delta\leq 4$, then ${\rm fas}(D)\leq m/3$. Both bounds are tight. Furthermore, the bound of (ii) is tight for degree-$4$ oriented multigraphs.

Figures (2)

  • Figure 1: Illustration of one of the cases in the proof of Theorem \ref{['thm:main']}.
  • Figure 2: Illustration of the different cases in Lemma \ref{['lem:5-regular']}. The thick arcs denote backward arcs.

Theorems & Definitions (21)

  • Theorem 1
  • Conjecture 1
  • Theorem 2
  • Conjecture 2
  • Theorem 3
  • Proposition 4
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 11 more