Asymptotic study of subcritical graph classes

International audienceWe present a unified general method for the asymptotic study of graphs from the so-called subcritical graph classes, which include the classes of cacti graphs, outerplanar graphs, and series-parallel graphs. This general method works both in the labelled and unlabelled framework. The main results concern the asymptotic enumeration and the limit laws of properties of random graphs chosen from subcritical classes. We show that the number $g_n/n!$ (resp. $g_n$) of labelled (resp. unlabelled) graphs on $n$ vertices from a subcritical graph class {\cG}=\cup_n {\cG_n} satisfies asymptotically the universal behaviour $$g_n = c n^{-5/2} \gamma^n\ (1+o(1))$$ for computable constants $c,\gamma$, e.g. $\gamma\approx 9.38527$ for unlabelled series-parallel graphs, and that the number of vertices of degree $k$ ($k$ fixed) in a graph chosen uniformly at random from \cG_n, converges (after rescaling) to a normal law as $n\to\infty$

Graph Treewidth and Geometric Thickness Parameters

Consider a drawing of a graph $G$ in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of $G$, is the classical graph parameter "thickness". By restricting the edges to be straight, we obtain the "geometric thickness". By further restricting the vertices to be in convex position, we obtain the "book thickness". This paper studies the relationship between these parameters and treewidth. Our first main result states that for graphs of treewidth $k$, the maximum thickness and the maximum geometric thickness both equal $\lceil{k/2}\rceil$. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth $k$, the maximum book thickness equals $k$ if $k \leq 2$ and equals $k+1$ if $k \geq 3$. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215-221, 2001]. Analogous results are proved for outerthickness, arboricity, and star-arboricity.Comment: A preliminary version of this paper appeared in the "Proceedings of the 13th International Symposium on Graph Drawing" (GD '05), Lecture Notes in Computer Science 3843:129-140, Springer, 2006. The full version was published in Discrete & Computational Geometry 37(4):641-670, 2007. That version contained a false conjecture, which is corrected on page 26 of this versio

Asymptotic Analysis of Outerplanar Graphs with Subgraph Obstructions

Σε αυτή την εργασία μελετάμε τεχνικές συνδυαστικής μοντελοποίησης καιασυμπτωτικής απαρίθμησης στο πλαίσιο της αναλυτικής συνδυαστικής. Ο οδηγός μας μέσα σε αυτή την πληθώρα τεχνικών είναι η απαρίθμηση κάποιων τύπων επίπεδων γραφημάτων, όταν εξαιρούμε συγκεκριμένα υπογραφήματα. Για την ακρίβεια, μελετάμε εξωεπίπεδα γραφήματα όταν ο περιορισμός είναι ο αποκλεισμός κύκλων συγκεκριμένου μήκους. Κατασκευάζουμε συνδυαστικές περιγραφές για τις δισυνεκτικές συνιστώσες που εξαιρούν τους κύκλους ώστε να αποκτήσουμε ασυμπτωτικά αποτελέσματα για τη γενική κλάση των εξωεπίπεδων γραφημάτων υπό περιορισμό. Οι προκλήσεις είναι συνδυαστικής αλλά και υπολογιστικής φύσης, καθώς οι περιγραφές γίνονται πιο περίπλοκες όταν το μήκος των κύκλων μεγαλώνει και οι γεννήτριες συναρτήσεις δίνονται σε έμμεση μορφή. Η συνδυαστική γλώσσα που χρησιμοποιείται είναι η λεγόμενη Συμβολική Μέθοδος που έρχεται μαζί με αντίστοιχες αναλυτικές τεχνικές.Επίσης μελετάμε παραμέτρους γενικών εξωεπίπεδων γραφημάτων, συγκεκριμένα τον αριθμό των τριγώνων και των τετραγώνων. Καταλήγουμε σε κανονικές οριακές κατανομές και εξάγουμε συγκεκριμένες σταθερές για την μέση τιμή και τη διασπορά.In this thesis we study techniques for combinatorial specification and asymptotic enumeration in the context of analytic combinatorics. Our guide through this variety of techniques is the enumeration of certain types of planar graphs, under subgraph exclusion constraints. In particular, we examine outerplanar graphs where the constraint is the exclusion of cycles of certain length. We build specifications for the 2-connected components that exclude certain cycles in order to obtain asymptotics for the general constrainded outerplanar class. The challenges here are combinatorial as well as computational, as the specifications become more involved when the length of the excluded cycle grows and the generating functions obtained are in implicit form. The combinatorial language that we use is the so-called Symbolic Method that comes in hand with corresponding analytic techniques. Furthermore, we study certain parameters of general outerplanar graphs, namely the number of triangles and quadrangles. We obtain Gaussian limiting distibutions and extract explicit constants for the mean and variance

Reconstructing Geometric Structures from Combinatorial and Metric Information

In this dissertation, we address three reconstruction problems. First, we address the problem of reconstructing a Delaunay triangulation from a maximal planar graph. A maximal planar graph G is Delaunay realizable if there exists a realization of G as a Delaunay triangulation on the plane. Several classes of graphs with particular graph-theoretic properties are known to be Delaunay realizable. One such class of graphs is outerplanar graph. In this dissertation, we present a new proof that an outerplanar graph is Delaunay realizable. Given a convex polyhedron P and a point s on the surface (the source), the ridge tree or cut locus is a collection of points with multiple shortest paths from s on the surface of P. If we compute the shortest paths from s to all polyhedral vertices of P and cut the surface along these paths, we obtain a planar polygon called the shortest path star (sp-star) unfolding. It is known that for any convex polyhedron and a source point, the ridge tree is contained in the sp-star unfolding polygon [8]. Given a combinatorial structure of a ridge tree, we show how to construct the ridge tree and the sp-star unfolding in which it lies. In this process, we address several problems concerning the existence of sp-star unfoldings on specified source point sets. Finally, we introduce and study a new variant of the sp-star unfolding called (geodesic) star unfolding. In this unfolding, we cut the surface of the convex polyhedron along a set of non-crossing geodesics (not-necessarily the shortest). We study its properties and address its realization problem. Finally, we consider the following problem: given a geodesic star unfolding of some convex polyhedron and a source point, how can we derive the sp-star unfolding of the same polyhedron and the source point? We introduce a new algorithmic operation and perform experiments using that operation on a large number of geodesic star unfolding polygons. Experimental data provides strong evidence that the successive applications of this operation on geodesic star unfoldings will lead us to the sp-star unfolding

Oriented Spanners

Given a point set $P$ in the Euclidean plane and a parameter $t$, we define an \emph{oriented $t$-spanner} as an oriented subgraph of the complete bi-directed graph such that for every pair of points, the shortest cycle in $G$ through those points is at most a factor $t$ longer than the shortest oriented cycle in the complete bi-directed graph. We investigate the problem of computing sparse graphs with small oriented dilation. As we can show that minimising oriented dilation for a given number of edges is NP-hard in the plane, we first consider one-dimensional point sets. While obtaining a $1$-spanner in this setting is straightforward, already for five points such a spanner has no plane embedding with the leftmost and rightmost point on the outer face. This leads to restricting to oriented graphs with a one-page book embedding on the one-dimensional point set. For this case we present a dynamic program to compute the graph of minimum oriented dilation that runs in $O(n^8)$ time for $n$ points, and a greedy algorithm that computes a $5$-spanner in $O(n\log n)$ time. Expanding these results finally gives us a result for two-dimensional point sets: we prove that for convex point sets the greedy triangulation results in an oriented $O(1)$-spanner.Comment: conference version: ESA '2

08431 Abstracts Collection -- Moderately Exponential Time Algorithms

From $19/10/2008$ to $24/10/2008$, the Dagstuhl Seminar 08431 ``Moderately Exponential Time Algorithms \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available