Gráfok és algoritmusok = Graphs and algorithms

A kutatás az elvárt eredménnyel zárult: tekintélyes nemzetközi konferenciákon és pubikációkban hoztuk nyilvánosságra az eredményéket, ideértve a STOC, SIAM és IEEE kiadványokat is, valamint egy könyvet is. A publikációk száma a matematikában elég magas (74). Ez nemzetközi összehasonlításban is kiemelkedő mutató a támogatás összegére vetítve. A projektben megmutattuk, hogy a gráfelmelet és a diszkrét matematika eszköztára számos helyen jól alkalmazható, ilyen terület a nagysebességű kommunikációs hálózatok tervezése, ezekben igen gyors routerek létrehozása. Egy másik terület a biológiai nagymolekulákon definiált gráfok és geometriai struktúrák. | The research concluded with the awaited results: in good international conferences and journals we published 74 works, including STOC conference, SIAM conferences and journals and one of the best IEEE journal. This number is high above average in mathematics research. We showed in the project that the tools of graph theory and discrete mathematics can be well applied in the high-speed communication network design, where we proposed fast and secure routing solutions. Additionally we also found applications in biological macromolecules

Optimal partisan districting on planar geographies

We show that optimal partisan districting in the plane with geographical constraints is an NP-complete problem

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 $\Delta-1$ edge slopes suffice for every outerplanar graph with maximum degree $\Delta\ge 4$. This improves on the previous bound of $O(\Delta^5)$, which was shown for planar partial 3-trees, a superclass of outerplanar graphs. The bound is tight: for every $\Delta\ge 4$ there is an outerplanar graph with maximum degree $\Delta$ that requires at least $\Delta-1$ distinct edge slopes in an outerplanar straight-line drawing.Comment: Major revision of the whole pape

Planar Octilinear Drawings with One Bend Per Edge

In octilinear drawings of planar graphs, every edge is drawn as an alternating sequence of horizontal, vertical and diagonal ($45^\circ$) line-segments. In this paper, we study octilinear drawings of low edge complexity, i.e., with few bends per edge. A $k$-planar graph is a planar graph in which each vertex has degree less or equal to $k$. In particular, we prove that every 4-planar graph admits a planar octilinear drawing with at most one bend per edge on an integer grid of size $O(n^2) \times O(n)$. For 5-planar graphs, we prove that one bend per edge still suffices in order to construct planar octilinear drawings, but in super-polynomial area. However, for 6-planar graphs we give a class of graphs whose planar octilinear drawings require at least two bends per edge

Decomposition of Geometric Set Systems and Graphs

We study two decomposition problems in combinatorial geometry. The first part deals with the decomposition of multiple coverings of the plane. We say that a planar set is cover-decomposable if there is a constant m such that any m-fold covering of the plane with its translates is decomposable into two disjoint coverings of the whole plane. Pach conjectured that every convex set is cover-decomposable. We verify his conjecture for polygons. Moreover, if m is large enough, we prove that any m-fold covering can even be decomposed into k coverings. Then we show that the situation is exactly the opposite in 3 dimensions, for any polyhedron and any $m$ we construct an m-fold covering of the space that is not decomposable. We also give constructions that show that concave polygons are usually not cover-decomposable. We start the first part with a detailed survey of all results on the cover-decomposability of polygons. The second part investigates another geometric partition problem, related to planar representation of graphs. The slope number of a graph G is the smallest number s with the property that G has a straight-line drawing with edges of at most s distinct slopes and with no bends. We examine the slope number of bounded degree graphs. Our main results are that if the maximum degree is at least 5, then the slope number tends to infinity as the number of vertices grows but every graph with maximum degree at most 3 can be embedded with only five slopes. We also prove that such an embedding exists for the related notion called slope parameter. Finally, we study the planar slope number, defined only for planar graphs as the smallest number s with the property that the graph has a straight-line drawing in the plane without any crossings such that the edges are segments of only s distinct slopes. We show that the planar slope number of planar graphs with bounded degree is bounded.Comment: This is my PhD thesi

Diszkrét és kombinatórikus geometriai kutatások = Topics in discrete and combinatorial geometry

A most lezárult OTKA grant, 8 résztvevő diszkrét geometriai kutatását támogatta. Itt a témák ilusztrálására kiemelünk néhányat az elért 72 publikációból. 1. Jelentős eredmények születtek (8 cikk) gráfok síkba rajzolhatóságáról, például az úgynevezett metszési számról. 2. Többek között sikerült igazolni Katchalski és Lewis 20 éves sejtését, mely szerint diszjunkt egységkörökből álló rendszereknél ha bármely három körnek van közös metsző egyenese akkor van olyan egyenes, amely legfeljebb 2 kör kivételével valamennyit metsz. 3. Littlewood (1964) problémájaként ismert volt az a kérdés, hogy hány henger érintheti kölcsönösen egymást? Viszonylag alacsony felső korlátot találtunk és egy régóta ismert elhelyzés valótlanságát igazoltuk. 4. Többszörös fedések egyszerű fedésekre való szétbontását vizsgáltuk és értünk el lényeges előrelépést. 5. A Borsuk-féle darabolási problémanak azt a variánsát vizsgáltuk, amelyben a darabolást u. n. hengeres darabolásra korlátozták. 6. Bebizonyítottuk, hogy ''nem nagyon elnyúlt'' ellipszisek esetében a sík legritkább fedésének meghatározásánál el lehet tekinteni az u.n. nem-keresztezési feltételtől. 7. A sejtetthez nagyon közeli korlátot találtunk arra a problémára, hogy az n-dimenziós térben legfeljebb hány homotetikus konvex test helyezhető el úgy, hogy bármely kettő érintse egymást. | Discrete geometry in Hungary flourished since the sixties as a result of the work of László Fejes Tóth. The supported research of 8 participant also belongs to this area. Here we illustrate the achieved 72 publications by mentioning a few results. 1. Important theorems (8 papers) were proved concerning graph drawing. 2. Among others, a 20 year old problem of Katchalsky was proved, stating that in a packing of congruent circles, if any three has a common transversal, then there is a line, which avoids at most two of the circles. 3. Concerning a conjecture of Littlewood we found a small upper bound for the number of infinite cylinders which mutually touch each other. 4. We studied decomposability of multiple coverings into single coverings. 5. We studied that variant of the famous Borsuk problem where the partitions are restricted to cylindrical partitions. 6. We proved that in case of ellipses which are not ''too long'' at determining the thinnest covering one can omit the usually needed noncrossing condition. 7. A bound close to the conjectured bound was found concerning the number of n-dimensional homothetic convex solids which mutually touch each other

Distinct Distances in Graph Drawings

The \emph{distance-number} of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the distance-number of trees, graphs with no $K^-_4$-minor, complete bipartite graphs, complete graphs, and cartesian products. Our main results concern the distance-number of graphs with bounded degree. We prove that $n$-vertex graphs with bounded maximum degree and bounded treewidth have distance-number in $\mathcal{O}(\log n)$. To conclude such a logarithmic upper bound, both the degree and the treewidth need to be bounded. In particular, we construct graphs with treewidth 2 and polynomial distance-number. Similarly, we prove that there exist graphs with maximum degree 5 and arbitrarily large distance-number. Moreover, as $\Delta$ increases the existential lower bound on the distance-number of $\Delta$-regular graphs tends to $\Omega(n^{0.864138})$

Level-Planar Drawings with Few Slopes

We introduce and study level-planar straight-line drawings with a fixed number of slopes. For proper level graphs (all edges connect vertices of adjacent levels), we give an ( log$^{2}$ / log log )-time algorithm that either finds such a drawing or determines that no such drawing exists. Moreover, we consider the partial drawing extension problem, where we seek to extend an immutable drawing of a subgraph to a drawing of the whole graph, and the simultaneous drawing problem, which asks about the existence of drawings of two graphs whose restrictions to their shared subgraph coincide. We present ($^{4/3}$ log )-time and ($^{10/3}$ log )-time algorithms for these respective problems on proper level-planar graphs. We complement these positive results by showing that testing whether non-proper level graphs admit level-planar drawings with slopes is NP-hard even in restricted cases