17 research outputs found

    Gráfok és algoritmusok = Graphs and algorithms

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    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

    Outerplanar graph drawings with few slopes

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    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 Δ1\Delta-1 edge slopes suffice for every outerplanar graph with maximum degree Δ4\Delta\ge 4. This improves on the previous bound of O(Δ5)O(\Delta^5), which was shown for planar partial 3-trees, a superclass of outerplanar graphs. The bound is tight: for every Δ4\Delta\ge 4 there is an outerplanar graph with maximum degree Δ\Delta that requires at least Δ1\Delta-1 distinct edge slopes in an outerplanar straight-line drawing.Comment: Major revision of the whole pape

    Drawings of Planar Graphs with Few Slopes and Segments

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    We study straight-line drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected plane graph on nn vertices has a plane drawing with at most 5/2n{5/2}n segments and at most 2n2n slopes. We prove that every cubic 3-connected plane graph has a plane drawing with three slopes (and three bends on the outerface). In a companion paper, drawings of non-planar graphs with few slopes are also considered.Comment: This paper is submitted to a journal. A preliminary version appeared as "Really Straight Graph Drawings" in the Graph Drawing 2004 conference. See http://arxiv.org/math/0606446 for a companion pape

    Distinct Distances in Graph Drawings

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    The \emph{distance-number} of a graph GG is the minimum number of distinct edge-lengths over all straight-line drawings of GG in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the distance-number of trees, graphs with no K4K^-_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 nn-vertex graphs with bounded maximum degree and bounded treewidth have distance-number in O(logn)\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 Δ\Delta increases the existential lower bound on the distance-number of Δ\Delta-regular graphs tends to Ω(n0.864138)\Omega(n^{0.864138})

    Level-Planar Drawings with Few Slopes

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    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 ( log2^{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^{4/3} log )-time and (10/3^{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
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