Small Superpatterns for Dominance Drawing

We exploit the connection between dominance drawings of directed acyclic graphs and permutations, in both directions, to provide improved bounds on the size of universal point sets for certain types of dominance drawing and on superpatterns for certain natural classes of permutations. In particular we show that there exist universal point sets for dominance drawings of the Hasse diagrams of width-two partial orders of size O(n^{3/2}), universal point sets for dominance drawings of st-outerplanar graphs of size O(n\log n), and universal point sets for dominance drawings of directed trees of size O(n^2). We show that 321-avoiding permutations have superpatterns of size O(n^{3/2}), riffle permutations (321-, 2143-, and 2413-avoiding permutations) have superpatterns of size O(n), and the concatenations of sequences of riffles and their inverses have superpatterns of size O(n\log n). Our analysis includes a calculation of the leading constants in these bounds.Comment: ANALCO 2014, This version fixes an error in the leading constant of the 321-superpattern siz

Superpatterns and Universal Point Sets

An old open problem in graph drawing asks for the size of a universal point set, a set of points that can be used as vertices for straight-line drawings of all n-vertex planar graphs. We connect this problem to the theory of permutation patterns, where another open problem concerns the size of superpatterns, permutations that contain all patterns of a given size. We generalize superpatterns to classes of permutations determined by forbidden patterns, and we construct superpatterns of size n^2/4 + Theta(n) for the 213-avoiding permutations, half the size of known superpatterns for unconstrained permutations. We use our superpatterns to construct universal point sets of size n^2/4 - Theta(n), smaller than the previous bound by a 9/16 factor. We prove that every proper subclass of the 213-avoiding permutations has superpatterns of size O(n log^O(1) n), which we use to prove that the planar graphs of bounded pathwidth have near-linear universal point sets.Comment: GD 2013 special issue of JGA

Universal Layered Permutations

We establish an exact formula for the length of the shortest permutation containing all layered permutations of length n, proving a conjecture of Gray

Embedding Four-directional Paths on Convex Point Sets

A directed path whose edges are assigned labels "up", "down", "right", or "left" is called \emph{four-directional}, and \emph{three-directional} if at most three out of the four labels are used. A \emph{direction-consistent embedding} of an \mbox{$n$-vertex} four-directional path $P$ on a set $S$ of $n$ points in the plane is a straight-line drawing of $P$ where each vertex of $P$ is mapped to a distinct point of $S$ and every edge points to the direction specified by its label. We study planar direction-consistent embeddings of three- and four-directional paths and provide a complete picture of the problem for convex point sets.Comment: 11 pages, full conference version including all proof

Drawing Arrangement Graphs In Small Grids, Or How To Play Planarity

We describe a linear-time algorithm that finds a planar drawing of every graph of a simple line or pseudoline arrangement within a grid of area O(n^{7/6}). No known input causes our algorithm to use area \Omega(n^{1+\epsilon}) for any \epsilon>0; finding such an input would represent significant progress on the famous k-set problem from discrete geometry. Drawing line arrangement graphs is the main task in the Planarity puzzle.Comment: 12 pages, 8 figures. To appear at 21st Int. Symp. Graph Drawing, Bordeaux, 201

A logarithmic bound for simultaneous embeddings of planar graphs

A set $\mathcal{G}$ of planar graphs on the same number $n$ of vertices is called simultaneously embeddable if there exists a set $P$ of $n$ points in the plane such that every graph $G \in \mathcal{G}$ admits a (crossing-free) straight-line embedding with vertices placed at points of $P$. A well-known open problem from 2007 posed by Brass, Cenek, Duncan, Efrat, Erten, Ismailescu, Kobourov, Lubiw and Mitchell, asks whether for some $n$ there exists a set $\mathcal{G}$ consisting of two planar graphs on $n$ vertices that are not simultaneously embeddable. While this remains widely open, we give a short proof that for every $\varepsilon>0$ and sufficiently large $n$ there exists a collection of at most $(4+\varepsilon)\log_2(n)$ planar graphs on $n$ vertices which cannot be simultaneously embedded. This significantly improves the previous exponential bound of $O(n\cdot 4^{n/11})$ for the same problem which was recently established by Goenka, Semnani and Yip.Comment: note, 5 page