Crossing-critical edges and Kuratowski subgraphs of a graph

Širáň‬, Jozef

doi:10.1016/0095-8956(83)90064-3

Cited by 6 publications

(4 citation statements)

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“…That such graphs exist was already exhibited by Kochol [8], who noted without proof thatŠiráň's result may only be true for simple graphs. Closely investigatingŠiráň's proof, it establishes [12] the following: Theorem 1.2 (Theorem 2 in [12]). Let e with endvertices b and c be a crossing-critical edge of a graph G for which cr(G − e) ≤ 1.…”

Section: Introductionmentioning

confidence: 99%

“…Following [12], for a (possibly empty) subset S of vertices of G, a pair (H, K) of subgraphs of G is an S-decomposition, if (i) each edge of G belongs to precisely one of H, K and…”

Section: Lemma 24mentioning

confidence: 99%

“…The following statement was exhibited in the same year. Statement 1.1 (Theorem 2 in [12]). Let e be a crossing-critical edge of a graph G, for which cr(G − e) ≤ 1.…”

Section: Introductionmentioning

confidence: 99%

“…Any subdivision of K 3,3 or K 5 in G is called a Kuratowski subgraph of G and an edge e is a Kuratowski edge, if e belongs to a Kuratowski subgraph of G. Any edge of G which is not a Kuratowski edge, will be called a non-Kuratowski edge. Following [12], we call an edge e of G k-exceptional if e is k-crossing-critical and e is a non-Kuratowski edge. Note that the existence of a k-exceptional edge in G for k > 0 implies the existence of a Kuratowski subgraph, and hence that G is non-planar.…”

Section: Introductionmentioning

confidence: 99%

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Characterizing all graphs with 2-exceptional edges

Bokal

Leaños

2018

Ars Math. Contemp.

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Dirac and Shuster in 1954 exhibited a simple proof of Kuratowski theorem by showing that any 1-crossing-critical edge of G belongs to a Kuratowski subdivision of G. In 1983, Siráň extended this result to any 2-crossing-critical edge e with endvertices b and c of a graph G with crossing number at least two, whenever no two blocks of G − b − c contain all its vertices. Calling an edge f of G k-exceptional whenever f is k-crossing-critical and it does not belong to any Kuratowski subgraph of G, he showed that simple 3-connected graphs with k-exceptional edges exist for any k ≥ 6, and they exist even for arbitrarily large difference of cr(G) − cr(G − f ). In 1991, Kochol constructed such examples for any k ≥ 4, and commented thatŠiráň's result holds for any simple graph.Examining the case when two blocks contain all the vertices of G − b − c, we show that graphs with k-exceptional edges exist for any k ≥ 2, albeit not necessarily simple. We confirm that no such simple graphs with 2-exceptional edges exist by applying the techniques of the recent characterization of 2-crossing-critical graphs to explicitly describe the set of all graphs with 2-exceptional edges and noting they all contain parallel edges. In this context, the paper can be read as an accessible prelude to the characterization of 2-crossing-critical graphs.

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Section: Introductionmentioning

confidence: 99%

“…Following [12], for a (possibly empty) subset S of vertices of G, a pair (H, K) of subgraphs of G is an S-decomposition, if (i) each edge of G belongs to precisely one of H, K and…”

Section: Lemma 24mentioning

confidence: 99%

“…The following statement was exhibited in the same year. Statement 1.1 (Theorem 2 in [12]). Let e be a crossing-critical edge of a graph G, for which cr(G − e) ≤ 1.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Characterizing all graphs with 2-exceptional edges

Bokal

Leaños

2018

Ars Math. Contemp.

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Infinite families of crossing‐critical graphs with prescribed average degree and crossing number

Bokal

2010

Journal of Graph Theory

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Širan constructed infinite families of ▫$k$▫-crossing-critical graphs for every ▫$k ge 3$▫ and Kochol constructed such families of simple graphs for every ▫$k ge 2$▫. Richter and Thomassen argued that, for any given ▫$k ge 1$▫ and ▫$r ge 6$▫, there are only finitely many simple ▫$k$▫-crossing-critical graphs with minimum degree ▫$r$▫. Salazar observed that the same argument implies such a conclusion for simple ▫$k$▫-crossing-critical graphs of prescribed average degree ▫$r > 6$▫. He established existence of infinite families of simple ▫$k$▫-crossing-critical graphs with any prescribed rational average degree ▫$r in [4,6)$▫ for infinitely many ▫$k$▫ and asked about their existence for ▫$r in (3,4)$▫. The question was partially settled by Pinontoan and Richter, who answered it positively for ▫$r in (3frac{1}{2},4)$▫. The present contribution uses two new constructions of crossing critical simple graphs along with the one developed by Pinontoan and Richter to unify these results and to answer Salazar\u27s question by the following statement: for every rational number ▫$r in (3,6)$▫ there exists an integer ▫$N_r$▫, such that, for any ▫$k > N_r$▫, there exists an infinite family of simple 3-connected crossing-critical graphs with average degree ▫$r$▫ and crossing number ▫$k$▫. Moreover, a universal lower bound on ▫$k$▫ applies for rational numbers in any closed interval ▫$I subset (3,6)$▫

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