Abstract
A straight-line respectively, polyline drawing Γ of a planar graph G on a set L k of k parallel lines is a planar drawing that maps each vertex of G to a distinct point on L k and each edge of G to a straight line segment respectively, a polygonal chain with the bends on L k between its endpoints. The height of Γ is k, i.e., the number of lines used in the drawing. In this paper we compute new upper bounds on the height of polyline drawings of planar graphs using planar separators. Specifically, we show that every n-vertex planar graph with maximum degree Δ, having a simple cycle separator of size λ, admits a polyline drawing with height 4n/9+OλΔ, where the previously best known bound was 2n/3. Since $\lambda\in O\sqrt{n}$ , this implies the existence of a drawing of height at most 4n/9+on for any planar triangulation with $\Delta \in o\sqrt{n}$ . For n-vertex planar 3-trees, we compute straight-line drawings with height 4n/9+O1, which improves the previously best known upper bound of n/2. All these results can be viewed as an initial step towards compact drawings of planar triangulations via choosing a suitable embedding of the input graph.
Highlights
A polyline drawing of a planar graph G is a planar drawing of G such that each vertex of G is mapped to a distinct point in the Euclidean plane, and each edge is mapped to a polygonal chain between its endpoints
Let Lk = {l1, l2, . . . , lk} be a set of k horizontal lines such that for each i ≤ k, line li passes through the point (0, i)
A polyline drawing of G is called a polyline drawing on Lk if the vertices and bends of the drawing lie on the lines of Lk
Summary
A polyline drawing of a planar graph G is a planar drawing of G such that each vertex of G is mapped to a distinct point in the Euclidean plane, and each edge is mapped to a polygonal chain between its endpoints. State-of-the-art algorithms that compute straight-line drawings of n-vertex plane graphs on an (O(n) × 2n/3)-size grid imply an upper bound of 2n/3 on the height of straight-line drawings [5, 6]. In this paper we show that every n-vertex planar graph with maximum degree ∆, having a simple cycle separator of size λ, admits a drawing with height 4n/9 + O(λ∆), which is better than the previously best known bound of 2n/3 for any λ∆ ∈ o(n). This result is an outcome of a new application of the planar separator theorem [8]. The drawing we construct in this case is a straight-line drawing
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