Andrei Asinowski

Vertex Intersection Graphs of Paths on a Grid

We investigate the class of vertex intersection graphs of paths on a grid, and specifically consider the subclasses that are obtained when each path in the representation has at most k bends (turns). We call such a subclass the Bk-VPG graphs, kC 0. In chip manufacturing, circuit layout is modeled as paths (wires) on a grid, where it is natural to constrain the number of bends per wire for reasons of feasibility and to reduce the cost of the chip. If the number k of bends is not restricted, then the VPG graphs are equivalent to the well-known class of string graphs, namely, the intersection graphs of arbitrary curves in the plane. In the case of B0-VPG graphs, we observe that horizontal and vertical segments have strong Helly number 2, and thus the clique problem has polynomial-time complexity, given the path representation. The recognition and coloring problems for B0-VPG graphs, however, are NPcomplete. We give a 2-approximation algorithm for coloring B0-VPG graphs. Furthermore, we prove that triangle-free B0-VPG graphs are 4-colorable, and this is best possible. We present a hierarchy of VPG graphs relating them to other known families of graphs. The grid intersection graphs are shown to be equivalent to the bipartite B0-VPG graphs and the circle graphs are strictly contained in B1-VPG. We prove the strict containment of B0-VPG into B1-VPG, and we conjecture that, in general, this strict containment continues for all values of k. We present a graph which is not in B1-VPG. Planar graphs are known to be in the class of string graphs, and we prove here that planar graphs are B3-VPG graphs, although it is not known if this is best possible.

Coloring hypergraphs induced by dynamic point sets and bottomless rectangles

We consider a coloring problem on dynamic, one-dimensional point sets: points appearing and disappearing on a line at given times. We wish to color them with k colors so that at any time, any sequence of p(k) consecutive points, for some function p, contains at least one point of each color. We prove that no such function p(k) exists in general. However, in the restricted case in which points appear gradually, but never disappear, we give a coloring algorithm guaranteeing the property at any time with p(k)=3k−2. This can be interpreted as coloring point sets in ℝ2 with k colors such that any bottomless rectangle containing at least 3k−2 points contains at least one point of each color. Here a bottomless rectangle is an axis-aligned rectangle whose bottom edge is below the lowest point of the set. For this problem, we also prove a lower bound p(k)>ck, where c>1.67. Hence, for every k there exists a point set, every k-coloring of which is such that there exists a bottomless rectangle containing ck points and missing at least one of the k colors. Chen et al. (2009) proved that no such function p(k) exists in the case of general axis-aligned rectangles. Our result also complements recent results from Keszegh and Palvolgyi on cover-decomposability of octants (2011, 2012).

Dyck paths with coloured ascents

We introduce a notion of Dyck paths with coloured ascents. For several ways of colouring, when the set of colours is itself some class of lattice paths, we establish bijections between sets of such paths and other combinatorial structures, such as non-crossing trees, dissections of a convex polygon, etc. In some cases enumeration gives new expression for sequences enumerating these structures.

String graphs of k-bend paths on a grid

Abstract We investigate the class of vertex intersection graphs of paths on a grid, and specifically consider the subclasses that are obtained when each path in the representation has at most k bends (turns). We call such a subclass the B k -VPG graphs, k ⩾ 0 . We present a complete hierarchy of VPG graphs relating them to other known families of graphs. String graphs are equivalent to VPG graphs. The grid intersection graphs [S. Bellantoni, I. Ben-Arroyo Hartman, T. Przytycka, S. Whitesides, Grid intersection graphs and boxicity, Discrete Math. 114, (1993), 41–49; I. Ben-Arroyo Hartman, I. Newman, R. Ziv, On grid intersection graphs, Discrete Math. 87(1), (1991), 41–52] are shown to be equivalent to the bipartite B 0 -VPG graphs. Chordal B 0 -VPG graphs are shown to be exactly Strongly Chordal B 0 -VPG graphs. We prove the strict containment of B 0 -VPG and circle graphs into B 1 -VPG. Planar graphs are known to be in the class of string graphs, and we prove here that planar graphs are B 3 -VPG graphs. In the case of B 0 -VPG graphs, we observe that a set of horizontal and vertical segments have strong Helly number 2. We show that the coloring problem for B k -VPG graphs, for k ⩾ 0 , is NP-complete and give a 2-approximation algorithm for coloring B 0 -VPG graphs. Furthermore, we prove that triangle-free B 0 -VPG graphs are 4-colorable, and this is best possible.

Point Sets with Many Non-Crossing Perfect Matchings

Abstract The maximum number of non-crossing straight-line perfect matchings that a set of n points in the plane can have is known to be O ( 10.0438 n ) and Ω ⁎ ( 3 n ) . The lower bound, due to Garcia, Noy, and Tejel (2000), is attained by the double chain, which has Θ ( 3 n / n Θ ( 1 ) ) such matchings. We reprove this bound in a simplified way that uses the novel notion of down-free matchings. We then apply this approach to several other constructions. As a result, we improve the lower bound. First we show that the double zigzag chain with n points has Θ ⁎ ( λ n ) non-crossing perfect matchings with λ ≈ 3.0532 . Next we analyze further generalizations of double zigzag chains – double r-chains. The best choice of parameters leads to a construction that has Θ ⁎ ( ν n ) matchings with ν ≈ 3.0930 . The derivation of this bound requires an analysis of a coupled dynamic-programming recursion between two infinite vectors.

Suballowable sequences and geometric permutations

We define suballowable sequences of permutations as a generalization of allowable sequences. We give a characterization of allowable sequences in the class of suballowable sequences, prove a Helly-type result on sets of permutations which form suballowable sequences, and show how suballowable sequences are related to problems of geometric realizability. We discuss configurations of points and geometric permutations in the plane. In particular, we find a characterization of pairwise realizability of planar geometric permutations, give two necessary conditions for realizability of planar geometric permutations, and show that these conditions are not sufficient.

The triples of geometric permutations for families of disjoint translates

Abstract A line meeting a family of pairwise disjoint convex sets induces two permutations of the sets. This pair of permutations is called a geometric permutation. We characterize the possible triples of geometric permutations for a family of disjoint translates in the plane. Together with earlier studies of geometric permutations this provides a complete characterization of realizable geometric permutations for disjoint translates.

Proper n-cell polycubes in n - 3 dimensions

A d-dimensional polycube of size n is a connected set of n cubes in d dimensions, where connectivity is through (d-1)-dimensional faces. Enumeration of polycubes, and, in particular, specific types of polycubes, as well as computing the asymptotic growth rate of polycubes, is a popular problem in discrete geometry. This is also an important tool in statistical physics for computations related to percolation processes and branched polymers. In this paper we consider proper polycubes: A polycube is said to be proper in d dimensions if the convex hull of the centers of its cubes is d-dimensional. We prove a formula for the number of polycubes of size n that are proper in (n - 3) dimensions.

Geometric Permutations of Large Families of Translates

Let F be a finite family of disjoint translates of a compact convex set K in R 2, and let l be an ordered line meeting each of the sets. Then l induces in the obvious way a total order on F. It is known that, up to reversals, at most three different orders can be induced on a given F as l varies. It is also known that the families are of six different types, according to the number of orders and their interrelations. In this paper we study these types closely, focusing on their relations to the given set K, and on what happens as |F| →∞.

The Maximal Number of Geometric Permutations for n Disjoint Translates of a Convex Set in ź Is Ω(n)

A geometric permutation induced by a transversal line of a finite family of disjoint convex sets in ℝd is the order in which the transversal meets the members of the family. It is known that the maximal number of geometric permutations in families of n disjoint translates of a convex set in ℝ3 is 3. We prove that for d ≥ 3 the maximal number of geometric permutations for such families in ℝd is Ω(n).