# Graph treewidth and geometric thickness parameters

Vida Dujmovic, David R. Wood

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5 Citations (Scopus)

### 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 thickness θ(G). By restricting the edges to be straight, we obtain the geometric thickness θ̄(G). By further restricting the vertices to be in convex position, we obtain the book thickness bt(G). This paper studies the relationship between these parameters and the treewidth of G. Let θ(Τk) / θ̄(Τ k) / bt(Τk) denote the maximum thickness / geometric thickness / book thickness of a graph with treewidth at most k. We prove that: - θ(Τk) = θ̄(Τk) = [k/2], and - bt(Τk) = k for k ≤ 2, and bt(Τk) = 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 bt(Τk) = k for all k. Analogous results are proved for outerthickness, arboricity, and star-arboricity.
Original language English Graph drawing Springer 129-140 12 3843 https://doi.org/10.1007/11618058_13 Published - 2006 Yes

### Publication series

Name Lecture Notes in Comput. Sci. Springer, Berlin

## Cite this

Dujmovic, V., & Wood, D. R. (2006). Graph treewidth and geometric thickness parameters. In Graph drawing (Vol. 3843, pp. 129-140). (Lecture Notes in Comput. Sci.). Springer. https://doi.org/10.1007/11618058_13