Thickness and Colorability of Geometric Graphs
The geometric thickness \(\bar\theta\)(G) of a graph G is the smallest integer t such that there exist a straight-line drawing Γ of G and a partition of its straight-line edges into t subsets, where each subset induces a planar drawing in Γ. Over a decade ago, Hutchinson, Shermer, and Vince proved that any n-vertex graph with geometric thickness two can have at most 6n − 18 edges, and for every n ≥ 8 they constructed a geometric thickness two graph with 6n − 20 edges. In this paper, we construct geometric thickness two graphs with 6n − 19 edges for every n ≥ 9, which improves the previously known 6n − 20 lower bound. We then construct a thickness two graph with 10 vertices that has geometric thickness three, and prove that the problem of recognizing geometric thickness two graphs is NP-hard, answering two questions posed by Dillencourt, Eppstein and Hirschberg. Finally, we prove the NP-hardness of coloring graphs of geometric thickness t with 4t − 1 colors, which strengthens a result of McGrae and Zito, when t = 2.
KeywordsPlanar Graph Complete Graph Planar Layer Geometric Graph Planar Drawing
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- 4.Barát, J., Matoušek, J., Wood, D.R.: Bounded-degree graphs have arbitrarily large geometric thickness. Electronic Journal of Combinatorics 13 (2006)Google Scholar
- 10.Duncan, C.A., Eppstein, D., Kobourov, S.G.: The geometric thickness of low degree graphs. In: Proc. of SoCG, pp. 340–346. ACM (2004)Google Scholar
- 16.Kainen, P.C.: Thickness and coarseness of graphs. Abhandlungen aus dem Mathematischen Seminar der Univ. Hamburg 39(88–95) (1973)Google Scholar
- 17.Mansfield, A.: Determining the thickness of a graph is NP-hard. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 93, pp. 9–23 (1983)Google Scholar
- 18.McGrae, A.R.A., Zito, M.: The complexity of the empire colouring problem. Algorithmica, 1–21 (2012) (published online)Google Scholar
- 20.Ringel, G.: Fabungsprobleme auf flachen und graphen. VEB Deutscher Verlag der Wissenschaften (1950)Google Scholar