140 research outputs found

    On the Obfuscation Complexity of Planar Graphs

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    Being motivated by John Tantalo's Planarity Game, we consider straight line plane drawings of a planar graph GG with edge crossings and wonder how obfuscated such drawings can be. We define obf(G)obf(G), the obfuscation complexity of GG, to be the maximum number of edge crossings in a drawing of GG. Relating obf(G)obf(G) to the distribution of vertex degrees in GG, we show an efficient way of constructing a drawing of GG with at least obf(G)/3obf(G)/3 edge crossings. We prove bounds (\delta(G)^2/24-o(1))n^2 < \obf G <3 n^2 for an nn-vertex planar graph GG with minimum vertex degree δ(G)≥2\delta(G)\ge 2. The shift complexity of GG, denoted by shift(G)shift(G), is the minimum number of vertex shifts sufficient to eliminate all edge crossings in an arbitrarily obfuscated drawing of GG (after shifting a vertex, all incident edges are supposed to be redrawn correspondingly). If δ(G)≥3\delta(G)\ge 3, then shift(G)shift(G) is linear in the number of vertices due to the known fact that the matching number of GG is linear. However, in the case δ(G)≥2\delta(G)\ge2 we notice that shift(G)shift(G) can be linear even if the matching number is bounded. As for computational complexity, we show that, given a drawing DD of a planar graph, it is NP-hard to find an optimum sequence of shifts making DD crossing-free.Comment: 12 pages, 1 figure. The proof of Theorem 3 is simplified. An overview of a related work is adde

    Applications of Graph Embedding in Mesh Untangling

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    The subject of this thesis is mesh untangling through graph embedding, a method of laying out graphs on a planar surface, using an algorithm based on the work of Fruchterman and Reingold[1]. Meshes are a variety of graph used to represent surfaces with a wide number of applications, particularly in simulation and modelling. In the process of simulation, simulated forces can tangle the mesh through deformation and stress. The goal of this thesis was to create a tool to untangle structured meshes of complicated shapes and surfaces, including meshes with holes or concave sides. The goals of graph embedding, such as minimizing edge crossings align very well with the objectives of mesh untangling. I have designed and tested a tool which I named MUT (Mesh Untangling Tool) on meshes of various types including triangular, polygonal, and hybrid meshes. Previous methods of mesh untangling have largely been numeric or optimizationbased. Additionally, most untangling methods produce low quality graphs which must be smoothed separately to produce good meshes. Currently graph embedding techniques have only been used for smoothing of untangled meshes. I have developed a tool based on the Fruchterman-Reingold algorithm for force-directed layout[1] that effectively untangles and smooths meshes simultaneously using graph embedding techniques. It can untangle complicated meshes with irregular polygonal frames, internal holes, and other complications that previous methods struggle with. The MUT does this by using several different approaches: untangling the mesh in stages from the frame in and anchoring the mesh at corner points to stabilize the untangling

    Untangling Circular Drawings: Algorithms and Complexity

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    We consider the problem of untangling a given (non-planar) straight-line circular drawing δG\delta_G of an outerplanar graph G=(V,E)G=(V, E) into a planar straight-line circular drawing by shifting a minimum number of vertices to a new position on the circle. For an outerplanar graph GG, it is clear that such a crossing-free circular drawing always exists and we define the circular shifting number shift(δG)(\delta_G) as the minimum number of vertices that are required to be shifted in order to resolve all crossings of δG\delta_G. We show that the problem Circular Untangling, asking whether shift(δG)≤K(\delta_G) \le K for a given integer KK, is NP-complete. For nn-vertex outerplanar graphs, we obtain a tight upper bound of shift(δG)≤n−⌊n−2⌋−2(\delta_G) \le n - \lfloor\sqrt{n-2}\rfloor -2. Based on these results we study Circular Untangling for almost-planar circular drawings, in which a single edge is involved in all the crossings. In this case, we provide a tight upper bound shift(δG)≤⌊n2⌋−1(\delta_G) \le \lfloor \frac{n}{2} \rfloor-1 and present a constructive polynomial-time algorithm to compute the circular shifting number of almost-planar drawings.Comment: 20 pages, 10 figures, extended version of ISAAC 2021 pape

    Separating sets of strings by finding matching patterns is almost always hard

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    © 2017 Elsevier B.V. We study the complexity of the problem of searching for a set of patterns that separate two given sets of strings. This problem has applications in a wide variety of areas, most notably in data mining, computational biology, and in understanding the complexity of genetic algorithms. We show that the basic problem of finding a small set of patterns that match one set of strings but do not match any string in a second set is difficult (NP-complete, W[2]-hard when parameterized by the size of the pattern set, and APX-hard). We then perform a detailed parameterized analysis of the problem, separating tractable and intractable variants. In particular we show that parameterizing by the size of pattern set and the number of strings, and the size of the alphabet and the number of strings give FPT results, amongst others

    Shortest path embeddings of graphs on surfaces

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    The classical theorem of F\'{a}ry states that every planar graph can be represented by an embedding in which every edge is represented by a straight line segment. We consider generalizations of F\'{a}ry's theorem to surfaces equipped with Riemannian metrics. In this setting, we require that every edge is drawn as a shortest path between its two endpoints and we call an embedding with this property a shortest path embedding. The main question addressed in this paper is whether given a closed surface S, there exists a Riemannian metric for which every topologically embeddable graph admits a shortest path embedding. This question is also motivated by various problems regarding crossing numbers on surfaces. We observe that the round metrics on the sphere and the projective plane have this property. We provide flat metrics on the torus and the Klein bottle which also have this property. Then we show that for the unit square flat metric on the Klein bottle there exists a graph without shortest path embeddings. We show, moreover, that for large g, there exist graphs G embeddable into the orientable surface of genus g, such that with large probability a random hyperbolic metric does not admit a shortest path embedding of G, where the probability measure is proportional to the Weil-Petersson volume on moduli space. Finally, we construct a hyperbolic metric on every orientable surface S of genus g, such that every graph embeddable into S can be embedded so that every edge is a concatenation of at most O(g) shortest paths.Comment: 22 pages, 11 figures: Version 3 is updated after comments of reviewer

    Non-crossing paths with fixed endpoints

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