Additivity of the crossing number of graphs with connectivity 2

Širáň‬, Jozef

doi:10.1007/bf02454162

Cited by 8 publications

(2 citation statements)

References 3 publications

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“…The proof of Theorem 3 is based on the following lemma which allows us to restrict the crossings in which the edges of a planar st-component may be involved so that we can still obtain a crossing minimal drawing of G. A similar result has been reported byŠiráň in [15]. However, as pointed out in [1], the proof given byŠiráň is not correct.…”

Section: Crossing Numbermentioning

confidence: 74%

Non-planar Core Reduction of Graphs

Gutwenger

Chimani

2006

Graph Drawing

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We present a reduction method that reduces a graph to a smaller core graph which behaves invariant with respect to planarity measures like crossing number, skewness, and thickness. The core reduction is based on the decomposition of a graph into its triconnected components and can be computed in linear time. It has applications in heuristic and exact optimization algorithms for the planarity measures mentioned above. Experimental results show that this strategy yields a reduction to 2/3 in average for a widely used benchmark set of graphs.

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Section: Crossing Numbermentioning

confidence: 74%

Non-planar Core Reduction of Graphs

Gutwenger

Chimani

2006

Graph Drawing

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“…We remark that the following discussion makes clear that the crossing number is not additive over cleavage units. Related discussions can be found in Širáň [32], Chimani, Gutwenger, and Mutzel [10] (but see [5] for significant comments about the latter), Beaudou and Bokal [5], and Leaños and Salazar [21]. Lemma 14.2.…”

Section: -Connected 2-critical Graphs That Are Not 3-connectedmentioning

confidence: 95%

Characterizing 2-crossing-critical graphs

Bokal¹,

Oporowski²,

Richter³

et al. 2013

Preprint

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List of Figures vChapter 1. Introduction Chapter 2. Description of 2-crossing-critical graphs with V 10 Chapter 3. Moving into the projective plane Chapter 4. Bridges Chapter 5. Quads have BOD Chapter 6. Green cycles Chapter 7. Exposed spoke with additional attachment not in Q 0 Chapter 8. G embeds with all spokes in M Chapter 9. Parallel edges Chapter 10. Tidiness and global H-bridges Chapter 11. Every rim edge has a colour Chapter 12. Existence of a red edge and its structure Chapter 13. The next red edge and the tile structure Chapter 14. Graphs that are not 3-connected 16.2. The number of bridges is bounded

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Crossing-critical edges and Kuratowski subgraphs of a graph

Širáň‬¹

1983

Journal of Combinatorial Theory, Series B

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Additivity of the crossing number of graphs with connectivity 2

Abstract: We prove that the crossing number of graphs with connectivity 2 has in certain cases an additive property analogous to that of crossing number of graphs with connectivity ~].

Cited by 8 publications

References 3 publications

Non-planar Core Reduction of Graphs

Non-planar Core Reduction of Graphs

Characterizing 2-crossing-critical graphs

Crossing-critical edges and Kuratowski subgraphs of a graph

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