Genus, Treewidth, and Local Crossing Number

Dujmović, Vida and Eppstein, David and Wood, David R. (2015) Genus, Treewidth, and Local Crossing Number. In: Graph Drawing and Network Visualization: 23rd International Symposium, GD 2015, September 24-26, 2015, Los Angeles, CA, USA , pp. 87-98 (Official URL:

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We consider relations between the size, treewidth, and local crossing number (maximum number of crossings per edge) of graphs embedded on topological surfaces. We show that an n-vertex graph embedded on a surface of genus g with at most k crossings per edge has treewidth O(sqrt((g+1)(k+1)n)) and layered treewidth O((g+1)k), and that these bounds are tight up to a constant factor. As a special case, the k-planar graphs with n vertices have treewidth O(sqrt((k+1)n)) and layered treewidth O(k+1), which are tight bounds that improve a previously known O((k+1)^(3/4)n^(1/2)) treewidth bound. Additionally, we show that for g<m, every m-edge graph can be embedded on a surface of genus g with O((m/(g+1))log^(2)g) crossings per edge, which is tight to a polylogarithmic factor.

Item Type:Conference Paper
Classifications:G Algorithms and Complexity > G.420 Crossings
Z Theory > Z.001 General
Z Theory > Z.750 Topology
ID Code:1480

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Dvorák, Z., Norin, S.: Treewidth of graphs with balanced separations. Electronic preprint arXiv: 1408.​3869 (2014)

Reed, B.A.: Tree width and tangles: a new connectivity measure and some applications. In: Bailey, R.A. (ed.) Surveys in Combinatorics. London Mathematical Society Lecture Note Series, vol. 241, pp. 87–162. Cambridge University Press, Cambridge (1997)

Schaefer, M.: The graph crossing number and its variants: a survey. Electron. J. Combin. DS21 (2014)

Grigoriev, A., Bodlaender, H.L.: Algorithms for graphs embeddable with few crossings per edge. Algorithmica 49(1), 1–11 (2007)

Guy, R. K., Jenkyns, T., Schaer, J.: The toroidal crossing number of the complete graph. J. Comb. Theor. 4, 376–390 (1968)

Gilbert, J.R., Hutchinson, J.P., Tarjan, R.E.: A separator theorem for graphs of bounded genus. J. Algorithms 5(3), 391–407 (1984)

Dujmović, V., Morin, P., Wood, D.R.: Layered separators in minor-closed families with applications. Electronic preprint arXiv: 1306.​1595 (2013)

Shahrokhi, F., Székely, L.A., Sýkora, O., Vrt’o, I.: Drawings of graphs on surfaces with few crossings. Algorithmica 16(1), 118–131 (1996)

Halin, R.: S-functions for graphs. J. Geometry 8(1–2), 171–186 (1976)

Robertson, N., Seymour, P.D.: Graph minors. II. algorithmic aspects of tree-width. J. Algorithms 7(3), 309–322 (1986)

Eppstein, D.: Diameter and treewidth in minor-closed graph families. Algorithmica 27, 275–291 (2000)

Hoory, S., Linial, N., Wigderson, A.: Expander graphs and their applications. Bull. Am. Math. Soc. 43(4), 439–561 (2006)

Grohe, M., Marx, D.: On tree width, bramble size, and expansion. J. Combin. Theory Ser. B 99(1), 218–228 (2009)

Leighton, T., Rao, S.: Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. J. ACM 46(6), 787–832 (1999)