Vít Jelínek

Upper bounds for the Stanley-Wilf limit of 1324 and other layered patterns

We prove that the Stanley-Wilf limit of any layered permutation pattern of length ? is at most 4?2, and that the Stanley-Wilf limit of the pattern 1324 is at most 16. These bounds follow from a more general result showing that a permutation avoiding a pattern of a special form is a merge of two permutations, each of which avoids a smaller pattern.We also conjecture that, for any k?0, the set of 1324-avoiding permutations with k inversions contains at least as many permutations of length n+1 as those of length n. We show that if this is true then the Stanley-Wilf limit for 1324 is at most eπ2/3?13.001954.

The planar slope number of planar partial 3-trees of bounded degree

It is known that every planar graph has a planar embedding where edges are represented by non-crossing straight-line segments. We study the planar slope number, i.e., the minimum number of distinct edge-slopes in such a drawing of a planar graph with maximum degree Δ. We show that the planar slope number of every series-parallel graph of maximum degree three is three. We also show that the planar slope number of every planar partial 3-tree and also every plane partial 3-tree is at most

Clustered Planarity: Embedded Clustered Graphs with Two-Component Clusters

2^{{\mathcal O}(\Delta)}

A Kuratowski-type theorem for planarity of partially embedded graphs

. In particular, we answer the question of Dujmovic et al. [Computational Geometry 38 (3), pp. 194–212 (2007)] whether there is a function f such that plane maximal outerplanar graphs can be drawn using at most f(Δ) slopes.

Dyck paths and pattern-avoiding matchings

We present a polynomial-time algorithm for c-planarity testing of clustered graphs with fixed plane embedding and such that every cluster induces a subgraph with at most two connected components.

Three optimal algorithms for balls of three colors

A partially embedded graph (or Peg) is a triple (G,H,H), where G is a graph, H is a subgraph of G, and H is a planar embedding of H. We say that a Peg(G,H,H) is planar if the graph G has a planar embedding that extends the embedding H. We introduce a containment relation of Pegs analogous to graph minor containment, and characterize the minimal non-planar Pegs with respect to this relation. We show that all the minimal non-planar Pegs except for finitely many belong to a single easily recognizable and explicitly described infinite family. We also describe a more complicated containment relation which only has a finite number of minimal non-planar Pegs. Furthermore, by extending an existing planarity test for Pegs, we obtain a polynomial-time algorithm which, for a given Peg, either produces a planar embedding or identifies an obstruction.

Clustered Planarity: Clusters with Few Outgoing Edges

How many matchings on the vertex set V = {1,2,...,2n} avoid a given configuration of three edges? Chen, Deng and Du have shown that the number of matchings that avoid three nesting edges is equal to the number of matchings avoiding three pairwise crossing edges. In this paper, we consider other forbidden configurations of size three. We present a bijection between matchings avoiding three crossing edges and matchings avoiding an edge nested below two crossing edges. This bijection uses non-crossing pairs of Dyck paths of length 2n as an intermediate step. Apart from that, we give a bijection that maps matchings avoiding two nested edges crossed by a third edge onto the matchings avoiding all configurations from an infinite family M, which contains the configuration consisting of three crossing edges. We use this bijection to show that for matchings of size n>3, it is easier to avoid three crossing edges than to avoid two nested edges crossed by a third edge. Our results on pattern-avoiding matchings can be regarded as an extension of previous results on pattern-avoiding permutations.

Counting general and self-dual interval orders

We consider a game played by two players, Paul and Carol. Carol fixes a coloring of n balls with three colors. At each step, Paul chooses a pair of balls and asks Carol whether the balls have the same color. Carol truthfully answers yes or no. In the Plurality problem, Paul wants to find a ball with the most common color. In the Partition problem, Paul wants to partition the balls according to their colors. He wants to ask Carol the least number of questions to reach his goal. We find optimal deterministic and probabilistic strategies for the Partition problem and an asymptotically optimal probabilistic strategy for the Plurality problem.

The Möbius function of separable and decomposable permutations

We present a linear algorithm for c-planarity testing of clustered graphs, in which every cluster has at most four outgoing edges.

Bend-bounded path intersection graphs: sausages, noodles, and waffles on a grill

In this paper, we present a new method to derive formulas for the generating functions of interval orders, counted with respect to their size, magnitude, and number of minimal and maximal elements. Our method allows us not only to generalize previous results on refined enumeration of general interval orders, but also to enumerate self-dual interval orders with respect to analogous statistics. Using the newly derived generating function formulas, we are able to prove a bijective relationship between self-dual interval orders and upper-triangular matrices with no zero rows. Previously, a similar bijective relationship has been established between general interval orders and upper-triangular matrices with no zero rows and columns.