Kolja Knauer

Convexity in Partial Cubes: The Hull Number

We prove that the combinatorial optimization problem of determining the hull number of a partial cube is NP-complete. This makes partial cubes the minimal graph class for which NP-completeness of this problem is known and improves earlier results in the literature. On the other hand we provide a polynomial-time algorithm to determine the hull number of planar partial cube quadrangulations. Instances of the hull number problem for partial cubes described include poset dimension and hitting sets for interiors of curves in the plane. To obtain the above results, we investigate convexity in partial cubes and obtain a new characterization of these graphs in terms of their lattice of convex subgraphs. This refines a theorem of Handa. Furthermore we provide a topological representation theorem for planar partial cubes, generalizing a result of Fukuda and Handa about tope graphs of rank 3 oriented matroids.

Reduction of interaction delays in networks

Delayed interactions are a common property of coupled natural systems and therefore arise in a variety of different applications. For instance, signals in neural or laser networks propagate at finite speed giving rise to delayed connections. Such systems are often modeled by delay differential equations with discrete delays. In realistic situations, these delays are not identical on different connections. We show that by a componentwise timeshift transformation it is often possible to reduce the number of different delays and simplify the models without loss of information. We identify dynamic invariants of this transformation, determine its capabilities to reduce the number of delays and interpret these findings in terms of the topology of the underlying graph. In particular, we show that networks with identical sums of delay times along the fundamental semicycles are dynamically equivalent and we provide a normal form for these systems. We illustrate the theory using a network motif of coupled Mackey-Glass systems with 8 different time delays, which can be reduced to an equivalent motif with three delays.

Uld-lattices and Δ-bonds

We provide a characterization of upper locally distributive lattices (ULD-lattices) in terms of edge colourings of their cover graphs. In many instances where a set of combinatorial objects carries the order structure of a lattice, this characterization yields a slick proof of distributivity or UL-distributivity. This is exemplified by proving a distributive lattice structure on Δ-bonds with invariant circular flow-difference. This instance generalizes several previously studied lattice structures, in particular, c-orientations (Propp), α-orientations of planar graphs (Felsner, resp. de Mendez) and planar flows (Khuller, Naor and Klein). The characterization also applies to other instances, e.g., to chip-firing games.

Outerplanar graph drawings with few slopes

We consider straight-line outerplanar drawings of outerplanar graphs in which a small number of distinct edge slopes are used, that is, the segments representing edges are parallel to a small number of directions. We prove that @D-1 edge slopes suffice for every outerplanar graph with maximum degree @D>=4. This improves on the previous bound of O(@D^5), which was shown for planar partial 3-trees, a superclass of outerplanar graphs. The bound is tight: for every @D>=4 there is an outerplanar graph with maximum degree @D that requires at least @D-1 distinct edge slopes in an outerplanar straight-line drawing.

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).

MAKING TRIANGLES COLORFUL

We prove that for any point set P in the plane, a triangle T, and a positive integer k, there exists a coloring of P with k colors such that any homothetic copy of T containing at least ck^8 points of P, for some constant c, contains at least one of each color. This is the first polynomial bound for range spaces induced by homothetic polygons. The only previously known bound for this problem applies to the more general case of octants in R^3, but is doubly exponential.

On the bend-number of planar and outerplanar graphs

The bend-numberb(G) of a graph G is the minimum k such that G may be represented as the edge intersection graph of a set of grid paths with at most k bends. We confirm a conjecture of Biedl and Stern showing that the maximum bend-number of outerplanar graphs is 2. Moreover we improve the formerly known lower and upper bound for the maximum bend-number of planar graphs from 2 and 5 to 3 and 4, respectively.

Distributive lattices, polyhedra, and generalized flows

A D-polyhedron is a polyhedron P such that if x,y are in P then so are their componentwise maximums and minimums. In other words, the point set of a D-polyhedron forms a distributive lattice with the dominance order. We provide a full characterization of the bounding hyperplanes of D-polyhedra. Aside from being a nice combination of geometric and order theoretic concepts, D-polyhedra are a unifying generalization of several distributive lattices which arise from graphs. In fact with a D-polyhedron we associate a directed graph with arc-parameters, such that points in the polyhedron correspond to vertex potentials on the graph. Alternatively, an edge-based description of the points of a D-polyhedron can be given. In this model the points correspond to the duals of generalized flows, i.e., duals of flows with gains and losses. These models can be specialized to yield distributive lattices that have been previously studied. Particular specializations are: flows of planar digraphs (Khuller, Naor and Klein), @a-orientations of planar graphs (Felsner), c-orientations (Propp) and @D-bonds of digraphs (Felsner and Knauer). As an additional application we identify a distributive lattice structure on generalized flow of breakeven planar digraphs.

How to eat 4/9 of a pizza

Two players want to eat a sliced pizza by alternately picking its pieces. The pieces may be of various sizes. After the first piece is eaten every subsequently picked piece must be adjacent to some previously eaten. We provide a strategy for the starting player to eat 49 of the total size of the pizza. This is best possible and settles a conjecture of Peter Winkler.

Intersection graphs of L-shapes and segments in the plane.

An L-shape is the union of a horizontal and a vertical segment with a common endpoint. These come in four rotations: \(\lfloor, \lceil, \rfloor\) and ⌉. A k-bend path is a simple path in the plane, whose direction changes k times from horizontal to vertical. If a graph admits an intersection representation in which every vertex is represented by an \(\lfloor\), an \(\lfloor\) or ⌈, a k-bend path, or a segment, then this graph is called an \(\lfloor\)-graph, \(\lfloor, \lceil\)-graph, B k -VPG-graph or SEG-graph, respectively. Motivated by a theorem of Middendorf and Pfeiffer [Discrete Mathematics, 108(1):365–372, 1992], stating that every \(\lfloor, \lceil\)-graph is a SEG-graph, we investigate several known subclasses of SEG-graphs and show that they are \(\lfloor\)-graphs, or B k -VPG-graphs for some small constant k. We show that all planar 3-trees, all line graphs of planar graphs, and all full subdivisions of planar graphs are \(\lfloor\)-graphs. Furthermore we show that all complements of planar graphs are B 19-VPG-graphs and all complements of full subdivisions are B 2-VPG-graphs. Here a full subdivision is a graph in which each edge is subdivided at least once.