The crossing numbers of products of path with graphs of order six

Klešč, Marián; Petrillová, Jana

doi:10.7151/dmgt.1684

Cited by 4 publications

(5 citation statements)

References 11 publications

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“…In 2013, Klešč and Petrillová [103] gave a summary of known results, including the crossing numbers of Cartesian products of path graphs with forty different graphs of order six. The majority of those results were first determined in [103] 111,121. These, along with the remaining decided cases are displayed in Table 5.…”

Section: Pathsmentioning

confidence: 99%

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A survey of graphs with known or bounded crossing numbers

Clancy¹,

Haythorpe²,

Newcombe³

2019

Preprint

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Section: Pathsmentioning

confidence: 99%

“…These, along with the remaining decided cases are displayed in Table 5. The authors responsible for deciding the cases outside of [103] are summarised in the following list. In total, the crossing number of G 6 i ✷P n has been decided for 58 of the 6-vertex graphs to date.…”

Section: Pathsmentioning

confidence: 99%

A survey of graphs with known or bounded crossing numbers

Clancy¹,

Haythorpe²,

Newcombe³

2019

Preprint

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“…[24]. There are only very few graph classes, e.g., Petersen graphs P (3, n) or Cartesian products of small graphs with paths or trees, see [4,21,25], for which the crossing number is known or can be efficiently computed. Considering approximations, we know that computing cr(G) is APX-hard [5], i.e., there does not exist a PTAS (unless P = NP).…”

Section: Introductionmentioning

confidence: 99%

Crossing Number for Graphs with Bounded Pathwidth

et al. 2020

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The crossing number is the smallest number of pairwise edge crossings when drawing a graph into the plane. There are only very few graph classes for which the exact crossing number is known or for which there at least exist constant approximation ratios. Furthermore, up to now, general crossing number computations have never been successfully tackled using bounded width of graph decompositions, like treewidth or pathwidth.In this paper, we for the first time show that crossing number is tractable (even in linear time) for maximal graphs of bounded pathwidth 3. The technique also shows that the crossing number and the rectilinear (a.k.a. straight-line) crossing number are identical for this graph class, and that we require only an O(n) × O(n)-grid to achieve such a drawing.Our techniques can further be extended to devise a 2-approximation for general graphs with pathwidth 3, and a 4w 3 -approximation for maximal graphs of pathwidth w. This is a constant approximation for bounded pathwidth graphs. ratios for special graph classes. In fact, all known constant approximation ratios are based on one of three concepts: Topology-based approximations require that G can be embedded without crossings on a surface of some fixed or bounded genus [14, 17, 18]. Insertion-based approximations assume that there is only a small (i.e., bounded size) subset of graph elements whose removal leaves a planar graph [6][7][8][9]. In either case, the ratios are constant only if we further assume bounded maximum degree. Finally, some approximations for the crossing number exist if the graph is dense [13].While treewidth and pathwidth have been very successful tools in many graph algorithm scenarios, they have only very rarely been applied to crossing number: Since general crossing number seems not to be describable with second order monadic logic, Courcelle's result [11] regarding treewidth-based tractability can only be applied if cr itself is bounded [15, 19]. The related strategy of "planar decompositions" lead to linear crossing number bounds [28].Contribution. In this paper, we for the first time show that such graph decompositions, in our case pathwidth, can be used for computing crossing number. We show for maximal graphs G of pathwidth 3 (see Section 3):We can compute the exact crossing number cr(G) in linear time.The topological cr(G) equals the rectilinear crossing number cr(G), i.e., the crossing number under the restriction that all edges need to be drawn as straight lines. We can compute a drawing realizing cr(G) on an O(n) × O(n)-grid.We then generalize these techniques to show: A 2-approximation for cr(G) and cr(G) for general graphs of pathwidth 3, see Section 4. A 4w 3 -approximation for cr(G) for maximal graphs of pathwidth w, see Section 5. This can be achieved by placing vertices and bend points on a 4n × wn grid. Observe that in contrast to most previous results, these approximation ratios are not dependent on the graph's maximum degree. As a complementary side note, we show (in the full version of the paper, see [1])...

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“…In the ensuing years, significant effort has gone into extending these results to include graphs on more vertices; in particular five and six vertices. The pioneering work in this area was by Klešč and his various co-authors [6][7][8][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31] who have spent the last three decades handling these cases, often on a graph-by-graph basis, requiring ad-hoc proofs that exploit the specific graph structure of the graphs in question. In the last fifteen years, a large number of other researchers have also contributed to this field.…”

Section: Introductionmentioning

confidence: 99%

“…Then, simply providing a drawing which establishes that cr(G 6 60 P n ) ≤ n−1 is sufficient to decide the cases for both G 6 46 P n and G 6 60 P n ; indeed, this exact argument was used in Klešč and Petrillová [28] to determine the crossing number of G In what follows, we use approaches similar to the previous paragraph to determine the crossing number for sixteen additional families of graphs. Although the arguments are not complicated, the extensive research into filling…”

Section: Introductionmentioning

confidence: 99%

On the Crossing Numbers of Cartesian Products of Small Graphs with Paths, Cycles and Stars

Clancy¹,

Haythorpe²,

Newcombe³

2019

Preprint

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There has been significant research dedicated towards computing the crossing numbers of families of graphs resulting from the Cartesian products of small graphs with arbitrarily large paths, cycles and stars. For graphs with four or fewer vertices, these have all been computed, but there are still various gaps for graphs with five or more vertices. We contribute to this field by determining the crossing numbers for sixteen such families.

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The crossing numbers of products of path with graphs of order six

Cited by 4 publications

References 11 publications

A survey of graphs with known or bounded crossing numbers

A survey of graphs with known or bounded crossing numbers

Crossing Number for Graphs with Bounded Pathwidth

On the Crossing Numbers of Cartesian Products of Small Graphs with Paths, Cycles and Stars

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