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

TL;DR

This paper introduces the fasd(D) parameter to study how many pairwise-disjoint feedback arc sets can partition a digraph's arcs, and analyzes both unweighted and weighted oriented graphs with bounded maximum degree and directed girth. It proves concrete bounds and exact values in key cases (notably fasd(4,3)=3 and fasd(3,g)=g for g=3,4,5), and develops a general framework showing fasd(Δ,g) is finite and often small, including a bound of 2 for large Δ via splitting and Ramanujan graphs. The authors employ a mix of constructive ordering techniques (good triples), combinatorial arguments, and spectral-expansion tools (Expander Mixing Lemma) to derive both upper and lower bounds, along with several explicit graph constructions that separate fas from fasd. The work advances understanding of Woodall-type conjectures in broader graph classes and pose several open problems for determining exact fasd values across more parameter regimes.

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 $Δ(D)$ 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 $Δ(D)\leq~4$ 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 $Δ(D)\leq 3$ and $g(D)\in \{3,4,5\}$, then ${\rm fasd}(D)=g(D)$; (iii) If $Δ(D)\leq 3$ and $g(D)\ge 8$ then ${\rm fasd}(D)<g(D).$ We also give some bounds for the cases when $Δ$ or $g$ are large and state several open problems and a conjecture.

Figures (9)

• Figure 1: The three cases of $P$. Bidirectional squiggly edges symbolize anti-directed paths.
• Figure 2: The subcases of Case 3. Bidirectional squiggly edges symbolize anti-directed paths.
• Figure 3: The three cases in the proof of Claim D when $l=3$. Dotted arcs can be absent.
• Figure 4: The coloring obtained in Claim G when (a) $w_1$ and $w_2$ are adjacent or (b) there exists a special coloring $c$ with $c(A^-(w_1)) = c(A^-(w_2))$.
• Figure 5: $D'$ and $D"$, when $D"$ is obtained from an undirected 5-cycle by replacing each edge with a directed 2-cycle. The dotted arcs indicate the arcs in $M$.

Theorems & Definitions (53)

• Theorem 1
• Corollary 1
• proof : Proof of Observation \ref{['obs']}
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• proof : Proof of Claim A