Chain Decompositions of 4-Connected Graphs

Curran, Sean; Lee, Orlando; Yu, Xingxing

doi:10.1137/s0895480103434592

Cited by 7 publications

(12 citation statements)

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“…Huck [5] proved the Independent Tree Conjecture for planar graphs (with any k). Building on this work and that of Kawarabayashi, Lee, and By adapting the technique of Schlipf and Schmidt [10], we prove an edge analog of the planar chain decomposition of Curran, Lee, and Yu [2]. We then use this decomposition to create two edge numberings which define the required trees.…”

Section: Introductionmentioning

confidence: 94%

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Four Edge-Independent Spanning Trees

Hoyer¹,

Thomas²

2018

SIAM J. Discrete Math.

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We prove an ear-decomposition theorem for 4-edge-connected graphs and use it to prove that for every 4-edge-connected graph G and every r ∈ V (G), there is a set of four spanning trees of G with the following property. For every vertex in G, the unique paths back to r in each tree are edge-disjoint. Our proof implies a polynomial-time algorithm for constructing the trees. Yu [7], the case k = 4 of the Independent Tree Conjecture was proven by Curran, Lee, and Yu across two papers [2, 3]. The Independent Tree Conjecture is open for nonplanar graphs with k > 4. In 1992, Khuller and Schieber [8] published a later-disproven argument that the Independent Tree Conjecture implies the Edge-Independent Tree Conjecture. Gopalan and Ramasubramanian [4] demonstrated that Khuller and Schieber's proof fails, but salvaged the technique, and proved the case k = 3 of the Edge-Independent Tree Conjecture by reducing it to the case k = 3 of the Independent Tree Conjecture. Schlipf and Schmidt [10] provided an alternate proof of the case k = 3 of the Edge-Independent Tree Conjecture, which does not rely on the Independent Tree Conjecture. The case k = 4 of the Edge-Independent Tree Conjecture is proven here, while the case k > 4 remains open.

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Section: Introductionmentioning

confidence: 94%

“…Throughout this section, fix a graph G with |V (G)| ≥ 1 and a vertex r ∈ V (G). We begin by defining a decomposition analogous to the planar chain decomposition in [2].…”

Section: The Chain Decompositionmentioning

confidence: 99%

Four Edge-Independent Spanning Trees

Hoyer¹,

Thomas²

2018

SIAM J. Discrete Math.

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“…This canonical ordering was extended to a canonical ordering for all planar 4-connected graphs (not necessarily triangulated) by Nakano, Rahman and Nishizeki [NRN97]. Versions of a canonical order for 4-connected non-planar graphs are also known [CLY05].…”

Section: Review Of Existing Canonical Orderingsmentioning

confidence: 99%

The (3,1)-ordering for 4-connected planar triangulations

Biedl¹,

Derka²

2016

JGAA

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Canonical orderings of planar graphs have frequently been used in graph drawing and other graph algorithms. In this paper we introduce the notion of an (r, s)-canonical order, which unifies many of the existing variants of canonical orderings. We then show that (3, 1)-canonical ordering for 4-connected triangulations always exist; to our knowledge this variant of canonical ordering was not previously known. We use it to give much simpler proofs of two previously known graph drawing results for 4-connected planar triangulations, namely, rectangular duals and rectangle-of-influence drawings. BackgroundA canonical ordering of a planar graph is a way of building the graph by iteratively attaching vertices to some "basic graph" (such as an edge) while preserving some connectivity invariant after each iteration. This concept was introduced in the late 1980's by de Fraysseix, Pach and Pollack [dFPP90]. They used the canonical ordering to show that planar graphs can be drawn on a grid of size (2n − 4) × (n − 2). Subsequently, canonical orderings became one of the main tools in graph drawings, e.g. for drawing graphs in grids of small dimensions (see e.g. [dFPP90, CN98]), rectangular duals [KH97], and also graph algorithms such as encoding planar graphs [HKL99] or finding k-disjoint trees in planar graphs [NRN97, NN00].Our contribution There is now a number of variations of canonical orderings, depending on the connectivity of the graph and whether it is triangulated or not. (We will review these below.) In this paper, we show the existence yet another canonical ordering, this one for planar 4-connected triangulations. It is substantially different from the canonical ordering for such graphs that was presented by Kant and He [KH97]. We call this the (3, 1)-canonical ordering. More generally, we introduce the concept of an (r, s)-canonical ordering, which (roughly speaking) means that the partial graph must be r-connected and the rest-graph must be s-connected; the existing canonical orders all fit into this framework.

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“…Chain decomposition [7] O(n 2 m), if planar: [23] O(m) 3 (3,1)-order [4] for triangulations O(m) 5-canonical decomposition [22] for triangulations O(m) 4 k\l 1 2 1…”

Section: Introductionmentioning

confidence: 99%

Edge-Orders

Schlipf

Schmidt

2018

Algorithmica

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Canonical orderings and their relatives such as st-numberings have been used as a key tool in algorithmic graph theory for the last decades. Recently, a unifying concept behind all these orders has been shown: they can be described by a graph decomposition into parts that have a prescribed vertex-connectivity.Despite extensive interest in canonical orderings, no analogue of this unifying concept is known for edge-connectivity. In this paper, we establish such a concept named edge-orders and show how to compute (1,1)-edge-orders of 2-edge-connected graphs as well as (2,1)-edge-orders of 3-edge-connected graphs in linear time, respectively. While the former can be seen as the edge-variants of st-numberings, the latter are the edge-variants of Mondshein sequences and non-separating ear decompositions. The methods that we use for obtaining such edge-orders differ considerably in almost all details from the ones used for their vertex-counterparts, as different graph-theoretic constructions are used in the inductive proof and standard reductions from edge-to vertex-connectivity are bound to fail.As a first application, we consider the famous Edge-Independent Spanning Tree Conjecture, which asserts that every k-edge-connected graph contains k rooted spanning trees that are pairwise edge-independent. We illustrate the impact of the above edge-orders by deducing algorithms that construct 2-and 3-edge independent spanning trees of 2-and 3-edge-connected graphs, the latter of which improves the best known running time from O(n 2 ) to linear time. * This research was supported by the DFG grant SCHM 3186/1-1. 1 arXiv:1607.04431v1 [cs.DM] 15 Jul 2016The purpose of this paper is to extent this unifying view further to (k, l)-edge-orders, in which prefixes are k-edge-connected and suffixes are l-edge-connected. Despite the many known and heavily used vertex-orders above, their natural edge-variants do not seem to be well-studied. In fact, we are only aware of one technical report by Annexstein et al. [1], which deals with (1,1)edge-orders (under the name st-edge-orderings). Besides this classification, we present a simple description how a (1,1)-edge-order can be computed. Our main contribution is then an algorithm that computes a (2,1)-edge-order of a 3-edge-connected graph in time O(m) (see Table 1 right), of which the corresponding result for the vertex-counterpart took over 40 years.From a top-level perspective, this latter result follows closely the proof outline used for its vertex-counterpart in [27]. However, each part of this proof requires new ideas and non-trivial formalizations: BG-sequences differ from Mader-sequences (and, although not too far apart, it took a 28-page paper to show that the latter can be computed in linear time as well [20]), both non-separateness and G i differ considerably already in their definitions, and, here, we need lastvalues in addition to just birth-values.Just like (2,1)-orders, which immediately led to improvements on the best-known running time for 5 applications [27,5], (2,1)-edge-ord...

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Chain Decompositions of 4-Connected Graphs

Cited by 7 publications

References 7 publications

Four Edge-Independent Spanning Trees

Four Edge-Independent Spanning Trees

The (3,1)-ordering for 4-connected planar triangulations

Edge-Orders

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