5,796 research outputs found

    A Quasi-Polynomial Time Partition Oracle for Graphs with an Excluded Minor

    Full text link
    Motivated by the problem of testing planarity and related properties, we study the problem of designing efficient {\em partition oracles}. A {\em partition oracle} is a procedure that, given access to the incidence lists representation of a bounded-degree graph G=(V,E)G= (V,E) and a parameter \eps, when queried on a vertex vVv\in V, returns the part (subset of vertices) which vv belongs to in a partition of all graph vertices. The partition should be such that all parts are small, each part is connected, and if the graph has certain properties, the total number of edges between parts is at most \eps |V|. In this work we give a partition oracle for graphs with excluded minors whose query complexity is quasi-polynomial in 1/\eps, thus improving on the result of Hassidim et al. ({\em Proceedings of FOCS 2009}) who gave a partition oracle with query complexity exponential in 1/\eps. This improvement implies corresponding improvements in the complexity of testing planarity and other properties that are characterized by excluded minors as well as sublinear-time approximation algorithms that work under the promise that the graph has an excluded minor.Comment: 13 pages, 1 figur

    Advances in C-Planarity Testing of Clustered Graphs

    Get PDF
    A clustered graph C=(G,T) consists of an undirected graph G and a rooted tree T in which the leaves of T correspond to the vertices of G=(V,E). Each vertex c in T corresponds to a subset of the vertices of the graph called ''cluster''. C-planarity is a natural extension of graph planarity for clustered graphs, and plays an important role in automatic graph drawing. The complexity status of c-planarity testing is unknown. It has been shown that c-planarity can be tested in linear time for c-connected graphs, i.e., graphs in which the cluster induced subgraphs are connected. In this paper, we provide a polynomial time algorithm for c-planarity testing for "almost" c-connected clustered graphs, i.e., graphs for which all c-vertices corresponding to the non-c-connected clusters lie on the same path in T starting at the root of T, or graphs in which for each non-connected cluster its super-cluster and all its siblings are connected. The algorithm uses ideas of the algorithm for subgraph induced planar connectivity augmentation. We regard it as a first step towards general c-planarity testing

    C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width

    Get PDF
    For a clustered graph, i.e, a graph whose vertex set is recursively partitioned into clusters, the C-Planarity Testing problem asks whether it is possible to find a planar embedding of the graph and a representation of each cluster as a region homeomorphic to a closed disk such that 1. the subgraph induced by each cluster is drawn in the interior of the corresponding disk, 2. each edge intersects any disk at most once, and 3. the nesting between clusters is reflected by the representation, i.e., child clusters are properly contained in their parent cluster. The computational complexity of this problem, whose study has been central to the theory of graph visualization since its introduction in 1995 [Qing-Wen Feng, Robert F. Cohen, and Peter Eades. Planarity for clustered graphs. ESA'95], has only been recently settled [Radoslav Fulek and Csaba D. T\'oth. Atomic Embeddability, Clustered Planarity, and Thickenability. To appear at SODA'20]. Before such a breakthrough, the complexity question was still unsolved even when the graph has a prescribed planar embedding, i.e, for embedded clustered graphs. We show that the C-Planarity Testing problem admits a single-exponential single-parameter FPT algorithm for embedded clustered graphs, when parameterized by the carving-width of the dual graph of the input. This is the first FPT algorithm for this long-standing open problem with respect to a single notable graph-width parameter. Moreover, in the general case, the polynomial dependency of our FPT algorithm is smaller than the one of the algorithm by Fulek and T\'oth. To further strengthen the relevance of this result, we show that the C-Planarity Testing problem retains its computational complexity when parameterized by several other graph-width parameters, which may potentially lead to faster algorithms.Comment: Extended version of the paper "C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width" to appear in the Proceedings of the 14th International Symposium on Parameterized and Exact Computation (IPEC 2019

    Výpočetní složitost testování rovinnosti grafu

    Get PDF
    In this paper we will show that the problem of planarity testing is in SL (symmetric nondeterministic LOGSPACE). The main part of our proof is a reduction of the problem to planarity of graphs with maximal degree three. Note that usual replacing vertices of degree bigger than three by "little circles" can spoil planarity, we need to be smarter. Planarity of graphs with maximal degree three was already solved in paper "Symmetric complementation" by John Reif. Previously Meena Mahajan and Eric Allender have already proved this in ("Complexity of planarity testing"), but their proof is the pure SL implementation of a parallel algorithm by John Reif and Vijaya Ramachandran ("Planarity testing in parallel"). But it is possibly unnecessarily complex and sophisticated for the purposes of the space complexity. This result together with recent breakthrough by Omer Reingold that SL = L ("Undirected T-connectivity in log-space") completely solves the question of complexity of planarity problem, because planarity is hard for L (it is again shown in "Complexity of planarity testing"). We construct logarithmic-space computable function that converts input graph G into G0 with maximal degree three such that G is planar if and only if G0 is. This together with.V tomto článku ukážeme, že testování planarity je v SL (symetrický nedeterministický LOGSPACE). Hlavní část našeho důkazu je redukce na problém testování rovinnosti grafu s maximálním stupněm tři. Povšiměte si, že obvyklé nahrazování vrchol větších stupňů "malými kružnicemi" může rovinnost pokazit, musíme si počínat šikovněji. Testování rovinnosti grafu s maximálním stupněm tři už bylo vyřešeno ve článku "Symmetric complementation" Johna Reifa. Už dříve Meena Mahajan a Eric Allender ("Complexity of planarity testing") ukázali, že testování rovinnosti je v SL. Jejich důkaz se však sestává z SL implementace velmi složitého paralelního algoritmu od Johna Reifa a Vijayi Ramachandran ("Planarity testing in parallel"). Ten je však nejspíše zbytečně komplikovaný pro účely prostorové složitosti. Tento výsledek spolu s nedávným průlomem Omera Reingolda dokazujícího, že SL = L ("Undirected ST-connectivity in log-space") zcela řeší otázku složitosti testování planarity, protože to je těžké pro L (toto je též dokázáno v "Complexity of planarity testing"). Zkonstruujeme algoritmus používající logaritmický prostor, který převede vstupní graf G na G0 s maximálním stupněm 3 tak, že G je rovinný tehdy a jen tehdy, když G0 je rovinný.Katedra aplikované matematikyDepartment of Applied MathematicsMatematicko-fyzikální fakultaFaculty of Mathematics and Physic

    Computing NodeTrix Representations of Clustered Graphs

    Full text link
    NodeTrix representations are a popular way to visualize clustered graphs; they represent clusters as adjacency matrices and inter-cluster edges as curves connecting the matrix boundaries. We study the complexity of constructing NodeTrix representations focusing on planarity testing problems, and we show several NP-completeness results and some polynomial-time algorithms. Building on such algorithms we develop a JavaScript library for NodeTrix representations aimed at reducing the crossings between edges incident to the same matrix.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

    Relaxing the Constraints of Clustered Planarity

    Full text link
    In a drawing of a clustered graph vertices and edges are drawn as points and curves, respectively, while clusters are represented by simple closed regions. A drawing of a clustered graph is c-planar if it has no edge-edge, edge-region, or region-region crossings. Determining the complexity of testing whether a clustered graph admits a c-planar drawing is a long-standing open problem in the Graph Drawing research area. An obvious necessary condition for c-planarity is the planarity of the graph underlying the clustered graph. However, such a condition is not sufficient and the consequences on the problem due to the requirement of not having edge-region and region-region crossings are not yet fully understood. In order to shed light on the c-planarity problem, we consider a relaxed version of it, where some kinds of crossings (either edge-edge, edge-region, or region-region) are allowed even if the underlying graph is planar. We investigate the relationships among the minimum number of edge-edge, edge-region, and region-region crossings for drawings of the same clustered graph. Also, we consider drawings in which only crossings of one kind are admitted. In this setting, we prove that drawings with only edge-edge or with only edge-region crossings always exist, while drawings with only region-region crossings may not. Further, we provide upper and lower bounds for the number of such crossings. Finally, we give a polynomial-time algorithm to test whether a drawing with only region-region crossings exist for biconnected graphs, hence identifying a first non-trivial necessary condition for c-planarity that can be tested in polynomial time for a noticeable class of graphs

    Computing k-Modal Embeddings of Planar Digraphs

    Get PDF
    Given a planar digraph G and a positive even integer k, an embedding of G in the plane is k-modal, if every vertex of G is incident to at most k pairs of consecutive edges with opposite orientations, i.e., the incoming and the outgoing edges at each vertex are grouped by the embedding into at most k sets of consecutive edges with the same orientation. In this paper, we study the k-Modality problem, which asks for the existence of a k-modal embedding of a planar digraph. This combinatorial problem is at the very core of a variety of constrained embedding questions for planar digraphs and flat clustered networks. First, since the 2-Modality problem can be easily solved in linear time, we consider the general k-Modality problem for any value of k>2 and show that the problem is NP-complete for planar digraphs of maximum degree Delta <= k+3. We relate its computational complexity to that of two notions of planarity for flat clustered networks: Planar Intersection-Link and Planar NodeTrix representations. This allows us to answer in the strongest possible way an open question by Di Giacomo [https://doi.org/10.1007/978-3-319-73915-1_37], concerning the complexity of constructing planar NodeTrix representations of flat clustered networks with small clusters, and to address a research question by Angelini et al. [https://doi.org/10.7155/jgaa.00437], concerning intersection-link representations based on geometric objects that determine complex arrangements. On the positive side, we provide a simple FPT algorithm for partial 2-trees of arbitrary degree, whose running time is exponential in k and linear in the input size. Second, motivated by the recently-introduced planar L-drawings of planar digraphs [https://doi.org/10.1007/978-3-319-73915-1_36], which require the computation of a 4-modal embedding, we focus our attention on k=4. On the algorithmic side, we show a complexity dichotomy for the 4-Modality problem with respect to Delta, by providing a linear-time algorithm for planar digraphs with Delta <= 6. This algorithmic result is based on decomposing the input digraph into its blocks via BC-trees and each of these blocks into its triconnected components via SPQR-trees. In particular, we are able to show that the constraints imposed on the embedding by the rigid triconnected components can be tackled by means of a small set of reduction rules and discover that the algorithmic core of the problem lies in special instances of NAESAT, which we prove to be always NAE-satisfiable - a result of independent interest that improves on Porschen et al. [https://doi.org/10.1007/978-3-540-24605-3_14]. Finally, on the combinatorial side, we consider outerplanar digraphs and show that any such a digraph always admits a k-modal embedding with k=4 and that this value of k is best possible for the digraphs in this family

    A Note on the Practicality of Maximal Planar Subgraph Algorithms

    Full text link
    Given a graph GG, the NP-hard Maximum Planar Subgraph problem (MPS) asks for a planar subgraph of GG with the maximum number of edges. There are several heuristic, approximative, and exact algorithms to tackle the problem, but---to the best of our knowledge---they have never been compared competitively in practice. We report on an exploratory study on the relative merits of the diverse approaches, focusing on practical runtime, solution quality, and implementation complexity. Surprisingly, a seemingly only theoretically strong approximation forms the building block of the strongest choice.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

    NodeTrix Planarity Testing with Small Clusters

    Full text link
    We study the NodeTrix planarity testing problem for flat clustered graphs when the maximum size of each cluster is bounded by a constant kk. We consider both the case when the sides of the matrices to which the edges are incident are fixed and the case when they can be chosen arbitrarily. We show that NodeTrix planarity testing with fixed sides can be solved in O(k3k+32n)O(k^{3k+\frac{3}{2}} \cdot n) time for every flat clustered graph that can be reduced to a partial 2-tree by collapsing its clusters into single vertices. In the general case, NodeTrix planarity testing with fixed sides can be solved in O(n)O(n) time for k=2k = 2, but it is NP-complete for any k>2k > 2. NodeTrix planarity testing remains NP-complete also in the free sides model when k>4k > 4.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017
    corecore