Feedback Arc Sets and Feedback Arc Set Decompositions in Weighted and Unweighted Oriented Graphs

TL;DR

This paper investigates bounds on feedback arc set decompositions and weights in directed graphs with constraints on degree and girth, contributing new bounds and confirming conjectures in specific cases.

Contribution

It provides new bounds on feedback arc set decomposition numbers and weights for oriented graphs with bounded degree and girth, extending known results and addressing open problems.

Findings

For max degree 4 and girth at least 3, the feedback arc set decomposition number is at least 3.

For max degree 3 and girth 3, 4, or 5, the decomposition number equals the girth.

For max degree 3 and girth at least 8, the decomposition number is less than the girth.

Abstract

Let $D=(V(D),A(D))$ be a digraph with at least one directed cycle. A set $F$ of arcs is a feedback arc set (FAS) if $D-F$ has no directed cycle. The FAS decomposition number ${\rm fasd}(D)$ of $D$ is the maximum number of pairwise disjoint FASs whose union is $A(D)$. The directed girth $g(D)$ of $D$ is the minimum length of a directed cycle of $D$. Note that ${\rm fasd}(D)\le g(D).$ The FAS decomposition number appears in the well-known and far-from-solved conjecture of Woodall (1978) stating that for every planar digraph $D$ with at least one directed cycle, ${\rm fasd}(D)=g(D).$ The degree of a vertex of $D$ is the sum of its in-degree and out-degree. Let $D$ be an arc-weighted digraph and let ${\rm fas}_w(D)$ denote the minimum weight of its FAS. In this paper, we study bounds on ${\rm fasd}(D)$, ${\rm fas}_w(D)$ and ${\rm fas}(D)$ for arc-weighted oriented graphs $D$ (i.e., digraphs without opposite arcs) with upper-bounded maximum degree $\Delta(D)$ and lower-bounded $g(D)$. Note that these parameters are related: ${\rm fas}_w(D)\le w(D)/{\rm fasd}(D)$, where $w(D)$ is the total weight of $D$, and ${\rm fas}(D)\le |A(D)|/{\rm fasd}(D).$ In particular, we prove the following: (i) If $\Delta(D)\leq~4$ and $g(D)\geq 3$, then ${\rm fasd}(D) \geq 3$ and therefore ${\rm fas}_w(D)\leq \frac{w(D)}{3}$ which generalizes a known tight bound for an unweighted oriented graph with maximum degree at most 4; (ii) If $\Delta(D)\leq 3$ and $g(D)\in \{3,4,5\}$, then ${\rm fasd}(D)=g(D)$; (iii) If $\Delta(D)\leq 3$ and $g(D)\ge 8$ then ${\rm fasd}(D)<g(D).$ We also give some bounds for the cases when $\Delta$ or $g$ are large and state several open problems and a conjecture.