Consider the following problem: Given a planar graph G, what is the maximum number p such that G has a planar straight-line drawing with p collinear vertices? This problem resides at the core of several graph drawing problems, including universal point subsets, untangling, and column planarity. The following results are known for it: Every n-vertex planar graph has a planar straight-line drawing with Ω( √ n) collinear vertices; for every n, there is an n-vertex planar graph whose every planar straight-line drawing has O(n σ ) collinear vertices, where σ < 0.986; every n-vertex planar graph of treewidth at most two has a planar straight-line drawing with Θ(n) collinear vertices. We extend the linear bound to planar graphs of treewidth at most three and to triconnected cubic planar graphs. This (partially) answers two open problems posed by Ravsky and Verbitsky [WG 2011:295-306]. Similar results are not possible for all bounded treewidth planar graphs or for all bounded degree planar graphs. For planar graphs of treewidth at most three, our results also imply asymptotically tight bounds for all of the other above mentioned graph drawing problems.
We study the problem of characterizing the directed graphs with an upward straight-line embedding into every point set in general or in convex position. We solve two questions posed by Binucci et al. [Computational Geometry: Theory and Applications, 2010]. Namely, we prove that the classes of directed graphs with an upward straight-line embedding into every point set in convex position and with an upward straight-line embedding into every point set in general position do not coincide, and we prove that every directed caterpillar admits an upward straight-line embedding into every point set in convex position. Further, we provide new partial positive results on the problem of constructing upward straight-line embeddings of directed paths into point sets in general position.
Abstract. A point set P ⊆ R 2 is universal for a class G if every graph of G has a planar straight-line embedding into P . We prove that there exists a O(n( log n log log n ) 2 ) size universal point set for the class of simply-nested n-vertex planar graphs. This is a step towards a full answer for the well-known open problem on the size of the smallest universal point sets for planar graphs [1,5,9].
A graph is called a strong ( resp. weak) bar 1-visibility graph if its vertices can be represented as horizontal segments (bars) in the plane so that its edges are all (resp. a subset of) the pairs of vertices whose bars have a -thick vertical line connecting them that intersects at most one other bar. We explore the relation among weak (resp. strong) bar 1-visibility graphs and other nearly planar graph classes. In particular, we study their relation to 1-planar graphs, which have a drawing with at most one crossing per edge; quasi-planar graphs, which have a drawing with no three mutually crossing edges; the squares of planar 1-flow networks, which are upward digraphs with in-or out-degree at most one. Our main results are that 1-planar graphs and the (undirected) squares of planar 1-flow networks are weak bar 1-visibility graphs and that these are quasi-planar graphs.
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