The main goal of this paper is to determine the asymptotic behavior of the number X_n of cut-vertices in random planar maps with n edges. It is shown that X_n/n → c in probability (for some explicit c>0). For so-called subcritial subclasses of planar maps like outerplanar maps we obtain a central limit theorem, too.

We show that for no surface except for the plane does monadic second-order logic (MSO) have a zero-one-law - and not even a convergence law - on the class of (connected) graphs embeddable on the surface. In addition we show that every rational in [0,1] is the limiting probability of some MSO formula. This strongly refutes a conjecture by Heinig et al. (2014) who proved a convergence law for planar graphs, and a zero-one law for connected planar graphs, and also identified the so-called gaps of [0,1]: the subintervals that are not limiting probabilities of any MSO formula. The proof relies on a combination of methods from structural graph theory, especially large face-width embeddings of graphs on surfaces, analytic combinatorics, and finite model theory, and several parts of the proof may be of independent interest. In particular, we identify precisely the properties that make the zero-one law work on planar graphs but fail for every other surface.

Let Pn denote a graph drawn uniformly at random from the class of all simple planar graphs with n vertices. We show that the maximum degree of a vertex in Pn is with probability 1 − o(1) asymptotically equal to c log n, where c ≈ 2.529 is determined explicitly. A similar result is also true for random 2-connected planar graphs. Our analysis combines two orthogonal methods that complement each other. First, in order to obtain the upper bound, we resort to exact methods, i.e., to generating functions and analytic combinatorics. This allows us to obtain fairly precise asymptotic estimates for the expected number of vertices of any given degree in Pn. On the other hand, for the lower bound we use Boltzmann sampling. In particular, by tracing the execution of an adequate algorithm that generates a random planar graph, we are able to explicitly find vertices of sufficiently high degree in Pn.

The Tutte polynomial is a notoriously hard graph invariant, and efficient algorithms for it are known only for a few special graph classes, like for those of bounded tree‐width. The notion of clique‐width extends the definition of cographs (graphs without induced $P_4$), and it is a more general notion than that of tree‐width. We show a subexponential algorithm (running in time $\exp{O(n^{1-\varepsilon})}\,$) for computing the Tutte polynomial on graphs of bounded clique‐width. In fact, our algorithm computes the more general U‐polynomial.

Abstract We present an automatic camera and mirrors placement method for visualizing 3D scenes such that complete and nonredundant visibility is guaranteed, as well as highest image resolution. We provide algorithms and geometries for optimal object‐independent mirrors and viewpoint placement for all objects in a specific class. We also briefly consider object‐dependent algorithmic problems, where mirrors and viewpoints are positioned as a function of the object.

Article Point-sets with few k-sets Share on Authors: Helmut Alt Institut für Informatik, Freie Universität Berlin, Takustr. 9, 14195 Berlin, Germany Institut für Informatik, Freie Universität Berlin, Takustr. 9, 14195 Berlin, GermanyView Profile , Stefan Felsner Institut für Informatik, Freie Universität Berlin, Takustr. 9 14195 Berlin, Germany Institut für Informatik, Freie Universität Berlin, Takustr. 9 14195 Berlin, GermanyView Profile , Ferran Hurtado Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Pau Gargallo 5, 08028-Barcelona, España Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Pau Gargallo 5, 08028-Barcelona, EspañaView Profile , Marc Noy Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Pau Gargallo 5, 08028-Barcelona, España Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Pau Gargallo 5, 08028-Barcelona, EspañaView Profile Authors Info & Claims SCG '98: Proceedings of the fourteenth annual symposium on Computational geometryJune 1998 Pages 200–205https://doi.org/10.1145/276884.276907Published:07 June 1998 0citation149DownloadsMetricsTotal Citations0Total Downloads149Last 12 Months0Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

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Article Flipping edges in triangulations Share on Authors: F. Hurtado Departamento de Matemática Aplicada II, Universitat Politécnica de Catalunya, Barcelona, Spain Departamento de Matemática Aplicada II, Universitat Politécnica de Catalunya, Barcelona, SpainView Profile , M. Noy Departamento de Matemática Aplicada II, Universitat Politécnica de Catalunya, Barcelona, Spain Departamento de Matemática Aplicada II, Universitat Politécnica de Catalunya, Barcelona, SpainView Profile , J. Urrutia Department of Computer Science, University of Ottawa, Ottawa, ON Canada Department of Computer Science, University of Ottawa, Ottawa, ON CanadaView Profile Authors Info & Claims SCG '96: Proceedings of the twelfth annual symposium on Computational geometryMay 1996 Pages 214–223https://doi.org/10.1145/237218.237367Online:01 May 1996Publication History 6citation419DownloadsMetricsTotal Citations6Total Downloads419Last 12 Months3Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access