GD 2006: Graph Drawing pp 150-161

# Planar Decompositions and the Crossing Number of Graphs with an Excluded Minor

• David R. Wood
• Jan Arne Telle
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4372)

## Abstract

Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar decomposition with bounded width and a linear number of bags.

The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. We prove that planar decompositions are intimately related to the crossing number, in the sense that a graph with bounded degree has linear crossing number if and only if it has a planar decomposition with bounded width and linear order. It follows from the above result about planar decompositions that every graph with bounded degree and an excluded minor has linear crossing number.

Analogous results are proved for the convex and rectilinear crossing numbers. In particular, every graph with bounded degree and bounded tree-width has linear convex crossing number, and every K 3,3-minor-free graph with bounded degree has linear rectilinear crossing number.

### References

1. 1.
Biedl, T., Demaine, E.D., Duncan, C.A., Fleischer, R., Kobourov, S.G.: Tight bounds on maximal and maximum matchings. Discrete Math. 285(1-3), 7–15 (2004)
2. 2.
Bienstock, D., Dean, N.: Bounds for rectilinear crossing numbers. J. Graph Theory 17(3), 333–348 (1993)
3. 3.
Bokal, D., Fijavž, G., Mohar, B.: The minor crossing number. SIAM J. Discrete Math. 20(2), 344–356 (2006)
4. 4.
Czabarka, É., Sýkora, O., Székely, L.A., Vrtó, I.: Outerplanar crossing numbers, the circular arrangement problem and isoperimetric functions. Electron. J. Combin. 11(1), R81 (2004)Google Scholar
5. 5.
Diestel, R.: Graph theory, 2nd edn. Graduate Texts in Mathematics, vol. 173. Springer, Heidelberg (2000)Google Scholar
6. 6.
Diestel, R., Kühn, D.: Graph minor hierarchies. Discrete Appl. Math. 145(2), 167–182 (2005)
7. 7.
Djidjev, H.N., [ERROR while converting LaTeX/Unicode], I.: Planar crossing numbers of genus g graphs. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 419–430. Springer, Heidelberg (2006)
8. 8.
Garcia-Moreno, E., Salazar, G.: Bounding the crossing number of a graph in terms of the crossing number of a minor with small maximum degree. J. Graph Theory 36(3), 168–173 (2001)
9. 9.
Leaños, J., Salazar, G.: On the additivity of crossing numbers of graphs (2006)Google Scholar
10. 10.
Mohar, B., Thomassen, C.: Graphs on surfaces. Johns Hopkins University Press, Baltimore (2001)
11. 11.
Pach, J., Tóth, G.: Crossing number of toroidal graphs. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 334–342. Springer, Heidelberg (2006)
12. 12.
Richter, R.B., Širáň, J.: The crossing number of K  3,n in a surface. J. Graph Theory 21(1), 51–54 (1996)
13. 13.
Robertson, N., Seymour, P.D.: Graph minors. XVI. Excluding a non-planar graph. J. Combin. Theory Ser. B 89(1), 43–76 (2003)
14. 14.
Scheinerman, E.R., Wilf, H.S.: The rectilinear crossing number of a complete graph and Sylvester’s “four point problem” of geometric probability. Amer. Math. Monthly 101(10), 939–943 (1994)
15. 15.
Shahrokhi, F., Sýkora, O., Székely, L.A., Vrt’o, I.: The gap between crossing numbers and convex crossing numbers. Towards a Theory of Geometric Graphs, vol. 342 of Contemporary Mathematics, pp. 249–258. Amer. Math. Soc (2004)Google Scholar
16. 16.
Székely, L.A.: A successful concept for measuring non-planarity of graphs: the crossing number. Discrete Math. 276(1-3), 331–352 (2004)
17. 17.
Wood, D.R., Telle, J.A.: Planar decompositions and the crossing number of graphs with an excluded minor (2006), http://www.arxiv.org/math/0604467

## Authors and Affiliations

• David R. Wood
• 1
• Jan Arne Telle
• 2
1. 1.Departament de Matemática Aplicada II, Universitat Politècnica de Catalunya, BarcelonaSpain
2. 2.Department of Informatics, The University of Bergen, BergenNorway