GD 2005: Graph Drawing pp 129-140

# Graph Treewidth and Geometric Thickness Parameters

• Vida Dujmović
• David R. Wood
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3843)

## Abstract

Consider a drawing of a graph G in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter thicknessθ(G). By restricting the edges to be straight, we obtain the geometric thickness$${\bar\theta}$$(G). By further restricting the vertices to be in convex position, we obtain the book thicknessbt(G). This paper studies the relationship between these parameters and the treewidth of G. Let $$\theta({\mathcal T}_{k}) / {\bar\theta}({\mathcal T}_{k}) / {\tt bt}({\mathcal T}_{k})$$ denote the maximum thickness / geometric thickness / book thickness of a graph with treewidth at most k. We prove that:

$$\theta({\mathcal T}_{k})={\bar\theta}({\mathcal T}_{k}) = \lceil k/2 \rceil$$, and

$${\tt bt}({\mathcal T}_{k}) = k$$ for k≤2, and $${\tt bt}({\mathcal T}_{k}) = k+1$$ for k≥3.

The first result says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. The second result disproves the conjecture of Ganley and Heath [Discrete Appl. Math. 2001] that $${\tt bt}({\mathcal T}_{k}) = k$$ for all k. Analogous results are proved for outerthickness, arboricity, and star-arboricity.

### References

1. 1.
Aoki, Y.: The star-arboricity of the complete regular multipartite graphs. Discrete Math. 81(2), 115–122 (1990)
2. 2.
Bernhart, F.R., Kainen, P.C.: The book thickness of a graph. J. Combin. Theory Ser. B 27(3), 320–331 (1979)
3. 3.
Di Giacomo, W.E., Didimo, G., Liotta, S.K.: Book embeddings and point-set embeddings of series-parallel digraphs. In: Goodrich, M.T., Kobourov, S.G. (eds.) GD 2002. LNCS, vol. 2528, pp. 162–173. Springer, Heidelberg (2002)
4. 4.
Ding, G., Oporowski, B., Sanders, D.P., Vertigan, D.: Partitioning graphs of bounded tree-width. Combinatorica 18(1), 1–12 (1998)
5. 5.
Dujmović, V., Wood, D.R.: On linear layouts of graphs. Discrete Math. Theor. Comput. Sci. 6(2), 339–358 (2004)
6. 6.
Dujmović, V., Wood, D.R.: Graph treewidth and geometric thickness parameters. arXiv.org:math.CO/0503553 (2005)Google Scholar
7. 7.
Ganley, J.L., Heath, L.S.: The pagenumber of k-trees is O(k). Discrete Appl. Math. 109(3), 215–221 (2001)
8. 8.
Guy, R.K.: Outerthickness and outercoarseness of graphs. In: Proc. British Combinatorial Conf. London Math. Soc. Lecture Note Ser., vol. 13, pp. 57–60. Cambridge Univ. Press, Cambridge (1974)Google Scholar
9. 9.
Halton, J.H.: On the thickness of graphs of given degree. Inform. Sci. 54(3), 219–238 (1991)
10. 10.
Kainen, P.C.: Thickness and coarseness of graphs. Abh. Math. Sem. Univ. Hamburg 39, 88–95 (1973)
11. 11.
Mutzel, P., Odenthal, T., Scharbrodt, M.: The thickness of graphs: a survey. Graphs Combin. 14(1), 59–73 (1998)
12. 12.
Nash-Williams, C.S.J.A.: Decomposition of finite graphs into forests. J. London Math. Soc. 39, 12 (1964)
13. 13.
Pach, J., Wenger, R.: Embedding planar graphs at fixed vertex locations. Graphs Combin. 17(4), 717–728 (2001)
14. 14.
Rengarajan, S., Veni Madhavan, C.E.: Stack and queue number of 2-trees. In: Li, M., Du, D.-Z. (eds.) COCOON 1995. LNCS, vol. 959, pp. 203–212. Springer, Heidelberg (1995)
15. 15.
Yannakakis, M.: Embedding planar graphs in four pages. J. Comput. System Sci. 38, 36–67 (1986)