On the Relationship Between k-Planar and k-Quasi-Planar Graphs

被引：2

作者：

Angelini, Patrizio ^{[1
]}

Bekos, Michael A. ^{[1
]}

Brandenburg, Franz J. ^{[2
]}

Da Lozzo, Giordano ^{[3
,4
]}

Di Battista, Giuseppe

Didimo, Walter ^{[5
]}

Liotta, Giuseppe ^{[5
]}

Montecchiani, Fabrizio ^{[5
]}

Rutter, Ignaz ^{[6
]}

机构：

[1] Univ Tubingen, Tubingen, Germany

[2] Univ Passau, Passau, Germany

[3] Univ Calif Irvine, Irvine, CA USA

[4] Roma Tre Univ, Rome, Italy

[5] Univ Perugia, Perugia, Italy

[6] TU Eindhoven, Eindhoven, Netherlands

来源：

GRAPH-THEORETIC CONCEPTS IN COMPUTER SCIENCE (WG 2017) | 2017年 / 10520卷

关键词：

CROSSING NUMBER; MAXIMUM NUMBER; EDGES; ALGORITHMS;

D O I：

10.1007/978-3-319-68705-6_5

中图分类号：

TP301 [理论、方法];

学科分类号：

081202 ;

摘要：

A graph is k-planar (k >= 1) if it can be drawn in the plane such that no edge is crossed k + 1 times or more. A graph is k-quasi-planar (k >= 2) if it can be drawn in the plane with no k pairwise crossing edges. The families of k-planar and k-quasi-planar graphs have been widely studied in the literature, and several bounds have been proven on their edge density. Nonetheless, only trivial results are known about the relationship between these two graph families. In this paper we prove that, for k >= 3, every k-planar graph is (k + 1)-quasi-planar.

引用

页码：59 / 74

页数：16

共 50 条

[1] k-Quasi-Planar Graphs
Suk, Andrew
[J]. GRAPH DRAWING, 2012, 7034 : 266 - 277
[2] THE NUMBER OF EDGES IN k-QUASI-PLANAR GRAPHS
Fox, Jacob
Pach, Janos
Suk, Andrew
[J]. SIAM JOURNAL ON DISCRETE MATHEMATICS, 2013, 27 (01) : 550 - 561
[3] On the Edge Density of k-Planar Graphs
Chaplick, Steven
Loeffler, Andre
Schmoeger, Rainer
[J]. GRAPH DRAWING AND NETWORK VISUALIZATION, GD 2018, 2018, 11282 : 636 - 639
[4] New bounds on the maximum number of edges in k-quasi-planar graphs
Suk, Andrew
Walczak, Bartosz
[J]. COMPUTATIONAL GEOMETRY-THEORY AND APPLICATIONS, 2015, 50 : 24 - 33
[5] Simple k-planar graphs are simple (k+1)-quasiplanar
Angelini, Patrizio
Bekos, Michael A.
Brandenburg, Franz J.
Da Lozzo, Giordano
Di Battista, Giuseppe
Didimo, Walter
Hoffmann, Michael
Liotta, Giuseppe
Montecchiani, Fabrizio
Rutter, Ignaz
Toth, Csaba D.
[J]. JOURNAL OF COMBINATORIAL THEORY SERIES B, 2020, 142 : 1 - 35
[6] k-planar crossing number of random graphs and random regular graphs
Asplund, John
Do, Thao
Hamm, Arran
Szekely, Laszlo
Taylor, Libby
Wang, Zhiyu
[J]. DISCRETE APPLIED MATHEMATICS, 2018, 247 : 419 - 422
[7] On k-planar crossing numbers
Shahrokhi, Farhad
Sykora, Ondrej
Szekely, Laszlo A.
Vrt'o, Imrich
[J]. DISCRETE APPLIED MATHEMATICS, 2007, 155 (09) : 1106 - 1115
[8] On the k-planar local crossing number
Asplund, John
Do, Thao
Hamm, Arran
Jain, Vishesh
[J]. DISCRETE MATHEMATICS, 2019, 342 (04) : 927 - 933
[9] Note on k-planar crossing numbers
Pach, Janos
Szekely, Laszlo A.
Toth, Csaba D.
Toth, Geza
[J]. COMPUTATIONAL GEOMETRY-THEORY AND APPLICATIONS, 2018, 68 : 2 - 6
[10] On Point Set Embeddings for k-Planar Graphs with Few Bends per Edge
Kaufmann, Michael
[J]. THEORY AND PRACTICE OF COMPUTER SCIENCE, SOFSEM 2019, 2019, 11376 : 260 - 271

← 1 2 3 4 5 →