On the Minimum Cut of Planarizations

Chimani, Markus; Gutwenger, Carsten; Mutzel, Petra

doi:10.1016/j.endm.2007.01.036

Cited by 4 publications

(6 citation statements)

References 6 publications

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“…In the second part, we describe the families of graphs that we use in the proof of our main result. In the last section, we point out a contradiction of our results with some arguments of Chimani, Gutwenger, and Mutzel [7]. These contradictions do not disprove their results, but only render their argument invalid.…”

Section: Introductioncontrasting

confidence: 75%

“…In [7], the authors claim to have proved that if C ⊆ E(G) is a minimum s, t-cut in a graph G and G s and G t are the components of G − C, then there exists an optimal drawing of G in which no edge of G s crosses an edge of G t . Since C is a minimum s, t-cut in G, G can be considered a zip product of graphs G x s and G y t , respectively obtained from G s and G t by adding a vertex x or y and connecting it to the endvertices of C in G s or G t .…”

Section: Two Families Of Graphsmentioning

confidence: 99%

Section: Two Families Of Graphsmentioning

confidence: 99%

“…Nevertheless, the paper [7] contains several original ideas, which could perhaps lead to a solution of Problems 4.6 or 4.7. Although our counterexample shows that the proof in [7] has a flaw, it does not disprove any of the main statements in that paper. These thus share the fate of the oldest result in the field of crossing numbers, the (still open) Zarankiewicz conjecture [11,27].…”

Section: Two Families Of Graphsmentioning

confidence: 99%

“…A crossing minimal planarization is a planarization, obtained from an optimal drawing. Problem 5.1 ( [7]) Let G be a connected graph and let s and t be two distinct vertices in G. Then there exists a crossing minimal planarization P of G, such that the size of the minimum s, t-cut in P is the same as the size of the minimum s, t-cut in G. Moreover, any crossing minimum planarization of G can be transformed into a crossing minimum planarization of G with the above property in linear time.…”

Section: Two Families Of Graphsmentioning

confidence: 99%

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On the Sharpness of Some Results Relating Cuts and Crossing Numbers

Beaudou

Bokal

2010

Electron. J. Combin.

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It is already known that for very small edge cuts in graphs, the crossing number of the graph is at least the sum of the crossing number of (slightly augmented) components resulting from the cut. Under stronger connectivity condition in each cut component that was formalized as a graph operation called zip product, a similar result was obtained for edge cuts of any size, and a natural question was asked, whether this stronger condition is necessary. In this paper, we prove that the relaxed condition is not sufficient when the size of the cut is at least four, and we prove that the gap can grow quadratically with the cut size.

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Section: Introductioncontrasting

confidence: 75%

Section: Two Families Of Graphsmentioning

confidence: 99%

Section: Two Families Of Graphsmentioning

confidence: 99%

Section: Two Families Of Graphsmentioning

confidence: 99%

Section: Two Families Of Graphsmentioning

confidence: 99%

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On the Sharpness of Some Results Relating Cuts and Crossing Numbers

Beaudou

Bokal

2010

Electron. J. Combin.

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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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Crossing number additivity over edge cuts

Bokal

Chimani

Leaños

2013

European Journal of Combinatorics

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We prove that if G is a graph with an minimal edge cut F of size three and G 1 , G 2 are the two (augmented) components of G − F , then the crossing number of G is equal to the sum of crossing numbers of G 1 and G 2 . Combining with known results, this implies that crossing number is additive over edge-cuts of size d for d ∈ {0, 1, 2, 3}, whereas there are counterexamples for every d ≥ 4. The techniques generalize to show that minor crossing number is additive over edge cuts of arbitrary size, as well as to provide bounds for crossing number additivity in arbitrary surfaces. We point out several applications to exact crossing number computation and crossing critical graphs, as well as provide a very general lower bound for the minor crossing number of the Cartesian product of an arbitrary graph with a tree.

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On the Minimum Cut of Planarizations

Cited by 4 publications

References 6 publications

On the Sharpness of Some Results Relating Cuts and Crossing Numbers

On the Sharpness of Some Results Relating Cuts and Crossing Numbers

Non-planar Core Reduction of Graphs

Crossing number additivity over edge cuts

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