Superpatterns and Universal Point Sets

Abstract

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 + Θ(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−Θ(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.

Universal point sets for planar three-trees

“…Angelini et al [1] generalized this result and showed that there exists an n-universal point set of size O (n(log n/ log log n) 2 ) for so-called simply nested planar graphs. A planar graph is simply nested if it can be reduced to an outerplanar graph by successively deleting chordless cycles from the boundary of the outer face.…”

“…A planar graph is simply nested if it can be reduced to an outerplanar graph by successively deleting chordless cycles from the boundary of the outer face. Recently, Bannister et al [2] found n-universal point sets of size O (n log n) for simply nested planar graphs, and O (n polylog n) for planar graphs of bounded pathwidth.…”

Universal Point Sets for Drawing Planar Graphs with Circular Arcs

“…Despite more than twenty years of research efforts, the best known lower bound for the value of f (n) is linear in n [6,18], while only an O(n 2 ) upper bound is known, as first established by de Fraysseix et al [7] and Schnyder [20]. Very recently, Bannister et al [2] showed a universal point set with n 2 /4 − Θ(n) points for planar straight-line drawings of n-vertex planar graphs. Universal point sets for planar straight-line drawings of planar graphs require more than n points whenever n 15 [5].…”

“…Universal point sets for planar straight-line drawings of planar graphs require more than n points whenever n 15 [5]. Universal point sets with o(n 2 ) points have been proved to exist for planar straight-line drawings of several subclasses of planar graphs, including simply-nested planar graphs [1,2], planar 3-trees [15], and graphs of bounded pathwidth [2].…”

On Universal Point Sets for Planar Graphs

“…On the one hand, the n-universal set given by De Fraysseix, Pach, and Pollack [16] has n 2 − O(n) points. This upper bound was improved by Schnyder [28] to n 2 /2 − O(n), then by Brandenburg [8] to 4n 2 /9 + O(n), and finally by Bannister et al [5] to n 2 /4 − Θ(n). On the other hand, Kurowski [23] showed that at least 1.235n points are necessary [23], improving earlier bounds of 1.206n by Chrobak and Karloff [14] and n + √ n by De Fraysseix, Pach, and Pollack [16].…”