Packing Dicycle Covers in Planar Graphs with No K 5–e Minor

Lee, Orlando; Williams, Aaron

doi:10.1007/11682462_62

Cited by 5 publications

(12 citation statements)

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“…There are several partial results restricting the attention to digraphs with certain properties. Lee and Wakabayashi [14] verified Conjecture 1 for digraphs whose underlying multigraph is series-parallel, which was later improved by Lee and Williams [15] proving the conjecture for digraphs whose underlying multigraph is planar and does not contain a triangular prism K 3 l K 2 as a minor * . Schrijver [20] and independently Feofiloff and Younger [9] verified the conjecture for source-sink connected digraphs, i.e.…”

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confidence: 89%

“…Although false in general, the conjecture of Edmonds and Giles has been verified in some special cases. In particular, the works by Lee and Wakabayashi [14], Lee and Williams [15], and Feofiloff and Younger [9] are actually about the conjecture of Edmonds and Giles and obtain corresponding results about Woodall's Conjecture as corollaries. For more research regarding this line of work, including a study of the structure of possible counterexamples, see [5,7,21,24].…”

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confidence: 97%

“…We turn our attention to infinite digraphs in Section 6. After showing that a more structural generalisation of Conjecture 1 to infinite digraphs fails (see Example 1), in Subsection 6.1 we use the compactness principle in combinatorics to finish the proof of Theorem 4 for infinite digraphs and prove a finitary version of the results of Lee and Williams [15] and Feofiloff and Younger [9]. Finally, in Subsection 6.2 we prove Theorem 5 and give an example that a capacitated version of that theorem fails.…”

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confidence: 98%

“…The parts of Theorems 3 and 4 regarding infinite digraphs are proved using the compactness principle in combinatorics. We will also use that technique to prove a finitary version of the results of Lee and Williams [15] and Feofiloff and Younger [9] for infinite digraphs only considering dicuts of finite size (capacity).…”

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confidence: 99%

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Disjoint dijoins for classes of dicuts in finite and infinite digraphs

Gollin¹,

Heuer²,

Stavropoulos³

2021

Preprint

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A dicut in a directed graph is a cut for which all of its edges are directed to a common side of the cut. A famous theorem of Lucchesi and Younger states that in every finite digraph the least size of a set of edges meeting every non-empty dicut equals the maximum number of disjoint dicuts in that digraph. Such sets are called dijoins. Woodall conjectured a dual statement. He asked whether the maximum number of disjoint dijoins in a directed graph equals the minimum size of a non-empty dicut.We study a modification of this question where we restrict our attention to certain classes of non-empty dicuts, i.e. whether for a class B of dicuts of a directed graph the maximum number of disjoint sets of edges meeting every dicut in B equals the size of a minimum dicut in B. In particular, we verify this questions for nested classes of finite dicuts, for the class of dicuts of minimum size, and for classes of infinite dibonds, and we investigate how this generalised setting relates to a capacitated version of this question. §1. IntroductionIn this paper we consider directed graphs, which we briefly denote as digraphs. A dicut in a digraph is a cut for which all of its edges are directed to a common side of the cut.A famous theorem of Lucchesi and Younger [16] states that in every finite digraph the least size of a set of edges meeting every non-empty dicut equals the maximum number of disjoint dicuts in that digraph. Such sets of edges are called dijoins.Woodall conjectured the following in some sense dual statement, where the roles of the minimum and maximum are reversed.Conjecture 1 (Woodall 1976 [25]). The size of a smallest non-empty dicut in a finite digraph D is equal to the size of the largest set of disjoint dijoins of D.

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confidence: 89%

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confidence: 97%

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confidence: 98%

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confidence: 99%

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Disjoint dijoins for classes of dicuts in finite and infinite digraphs

Gollin¹,

Heuer²,

Stavropoulos³

2021

Preprint

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“…[wt, τ ] is known to be true for instances (D, w) where the underlying undirected graph of D is series-parallel [28], or more generally has no K 5 \ e minor [29]. Edmonds and Giles [17] conjectured that [wt, τ ] is true, but Schrijver refuted the conjecture [36] by exhibiting a counterexample, others were also found later [9,41].…”

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confidence: 99%

On packing dijoins in digraphs and weighted digraphs

Abdi¹,

Cornuéjols²,

Zlatin³

2022

Preprint

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In this paper, we make some progress in addressing Woodall's Conjecture, and the refuted Edmonds-Giles Conjecture on packing dijoins in unweighted and weighted digraphs. Let D = (V, A) be a digraph, and let w ∈ Z A ≥0 . A dicut is a cut δ + (U ) ⊆ A for some nonempty proper vertex subset U such that δ − (U ) = ∅, a dijoin is an arc subset that intersects every dicut at least once, and more generally a k-dijoin is an arc subset that intersects every dicut at least k times.Suppose every dicut has weight at least τ , for some integer τ ≥ 2. Let ρ(τ, D, w) := 1 τ v∈V m v , where each m v is the integer in {0, 1, . . . , τ − 1} equal to w(δ + (v)) − w(δ − (v)) mod τ . In this paper, we prove the following results, amongst others:1. If w = 1, then A can be partitioned into a k-dijoin and a (τ − k)-dijoin, for every k ∈ [τ ].2. If ρ(τ, D, w) ∈ {0, 1}, then there is an equitable w-weighted packing of dijoins of size τ .3. If ρ(τ, D, w) = 2, then there is a w-weighted packing of dijoins of size τ . 4. If w = 1 and ρ(τ, D, w) = 3, then A can be partitioned into two dijoins and a (τ − 2)-dijoin.Each result is best possible: (1) and ( 4) do not hold for general w, (2) does not hold for ρ(τ, D, w) = 2 even if w = 1, and (3) does not hold for ρ(τ, D, w) = 3.The results are rendered possible by a Decompose, Lift, and Reduce procedure, which turns (D, w) into a set of sink-regular weighted (τ, τ +1)-bipartite digraphs, each of which is a weighted digraph where every vertex is a sink of weighted degree τ or a source of weighted degree τ, τ + 1, and every dicut has weight at least τ . From there, an application of the classical alternating path technique proves (2). For each of the weighted digraphs, we define two matroids M 0 , M 1 of rank ρ(τ, D, w) whose common ground set is the set of sources of weighted degree τ + 1, where M 0

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On Packing Dijoins in Digraphs and Weighted Digraphs

Abdi,

Cornuéjols,

Zlatin

2023

SIAM J. Discrete Math.

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Let D = (V, A) be a digraph. A dicut is a cut δ + (U ) ⊆ A for some nonempty proper vertex subset U such that δ − (U ) = ∅, a dijoin is an arc subset that intersects every dicut at least once, and more generally a k-dijoin is an arc subset that intersects every dicut at least k times. Our first result is that A can be partitioned into a dijoin and a (τ − 1)-dijoin where τ denotes the smallest size of a dicut. Woodall conjectured the stronger statement that A can be partitioned into τ dijoins.Let w ∈ Z A ≥0 and suppose every dicut has weight at least τ , for some integer τ ≥ 2. Let ρ(τ, D, w) := 1 τ v∈V m v , where each m v is the integer in {0, 1, . . . , τ − 1} equal to w(δ + (v)) − w(δ − (v)) mod τ . We prove the following results:(i) If ρ(τ, D, w) ∈ {0, 1}, then there is an equitable w-weighted packing of dijoins of size τ .(ii) If ρ(τ, D, w) = 2, then there is a w-weighted packing of dijoins of size τ .(iii) If ρ(τ, D, w) = 3, τ = 3, and w = 1, then A can be partitioned into three dijoins.Each result is best possible: (i) does not hold for ρ(τ, D, w) = 2 even if w = 1, (ii) does not hold for ρ(τ, D, w) = 3, and (iii) do not hold for general w.

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Packing Dicycle Covers in Planar Graphs with No K 5–e Minor

Cited by 5 publications

References 12 publications

Disjoint dijoins for classes of dicuts in finite and infinite digraphs

Disjoint dijoins for classes of dicuts in finite and infinite digraphs

On packing dijoins in digraphs and weighted digraphs

On Packing Dijoins in Digraphs and Weighted Digraphs

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