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Generating Labeled Planar Graphs Uniformly at Random
, 2003
"... We present an expected polynomial time algorithm to generate a labeled planar graph uniformly at random. To generate the planar graphs, we derive recurrence formulas that count all such graphs with n vertices and m edges, based on a decomposition into 1, 2, and 3connected components. For 3con ..."
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Cited by 30 (5 self)
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We present an expected polynomial time algorithm to generate a labeled planar graph uniformly at random. To generate the planar graphs, we derive recurrence formulas that count all such graphs with n vertices and m edges, based on a decomposition into 1, 2, and 3connected components. For 3connected graphs we apply a recent random generation algorithm by Schaeffer and a counting formula by Mullin and Schellenberg.
Enumeration and Asymptotic Properties of Unlabeled Outerplanar Graphs
 JOURNAL OF COMBINATORICS
, 2007
"... We determine the exact and asymptotic number of unlabeled outerplanar graphs. The exact number gn of unlabeled outerplanar graphs on n vertices can be computed in polynomial time, and gn is asymptotically g n −5/2 ρ −n, where g ≈ 0.00909941 and ρ −1 ≈ 7.50360 can be approximated. Using our enumerati ..."
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Cited by 7 (3 self)
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We determine the exact and asymptotic number of unlabeled outerplanar graphs. The exact number gn of unlabeled outerplanar graphs on n vertices can be computed in polynomial time, and gn is asymptotically g n −5/2 ρ −n, where g ≈ 0.00909941 and ρ −1 ≈ 7.50360 can be approximated. Using our enumerative results we investigate several statistical properties of random unlabeled outerplanar graphs on n vertices, for instance concerning connectedness, the chromatic number, and the number of edges. To obtain the results we combine classical cycle index enumeration with recent results from analytic combinatorics.
Sampling unlabeled biconnected planar graphs
 IN THE PROCEEDINGS OF THE 16TH ANNUAL INTERNATIONAL SYMPOSIUM ON ALGORITHMS AND COMPUTATION (ISAAC’05), 2005, SPRINGER LNCS 3827, 593 – 603
"... We present an expected polynomial time algorithm to generate a 2connected unlabeled planar graph uniformly at random. To do this we first derive recurrence formulas to count the exact number of rooted 2connected planar graphs, based on a decomposition along the connectivity structure. For 3conn ..."
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Cited by 4 (4 self)
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We present an expected polynomial time algorithm to generate a 2connected unlabeled planar graph uniformly at random. To do this we first derive recurrence formulas to count the exact number of rooted 2connected planar graphs, based on a decomposition along the connectivity structure. For 3connected planar graphs we use the fact that they have a unique embedding on the sphere. Special care has to be taken for rooted graphs that have a sensereversing or a poleexchanging automorphism. We prove a bijection between such symmetric objects and certain colored networks. These colored networks can again be decomposed along their connectivity structure. All the numbers can be evaluated in polynomial time by dynamic programming. To generate 2connected unlabeled planar graphs without a root uniformly at random we apply rejection sampling and obtain an expected polynomial time algorithm.
Random Cubic Planar Graphs
"... We show that the number of labeled cubic planar graphs on n vertices with n even is asymptotically αn −7/2 ρ −n n!, where ρ −1. = 3.13259 and α are analytic constants. We show also that the chromatic number of a random cubic planar graph that is chosen uniformly at random among all the labeled cubic ..."
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Cited by 4 (2 self)
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We show that the number of labeled cubic planar graphs on n vertices with n even is asymptotically αn −7/2 ρ −n n!, where ρ −1. = 3.13259 and α are analytic constants. We show also that the chromatic number of a random cubic planar graph that is chosen uniformly at random among all the labeled cubic planar graphs on n vertices is three with probability tending to e −ρ4 /4!. = 0.999568, and is four with probability tending to 1−e −ρ 4 /4! as n → ∞ with n even. The proof given combines generating function techniques with probabilistic arguments.
The Maximum Degree of Random Planar Graphs
"... 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 ..."
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Cited by 2 (0 self)
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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 2connected 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.
Generating Unlabeled Connected Cubic Planar Graphs Uniformly at Random
"... We present an expected polynomial time algorithm to generate an unlabeled connected cubic planar graph uniformly at random. We first consider rooted connected cubic planar graphs, i.e., we count connected cubic planar graphs up to isomorphisms that fix a certain directed edge. Based on decompositi ..."
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We present an expected polynomial time algorithm to generate an unlabeled connected cubic planar graph uniformly at random. We first consider rooted connected cubic planar graphs, i.e., we count connected cubic planar graphs up to isomorphisms that fix a certain directed edge. Based on decompositions along the connectivity structure, we derive recurrence formulas for the exact number of rooted cubic planar graphs. This leads to rooted 3connected cubic planar graphs, which have a unique embedding on the sphere. Special care has to be taken for rooted graphs that have a sensereversing automorphism. Therefore we introduce the concept of colored networks, which stand in bijective correspondence to rooted 3connected cubic planar graphs with given symmetries. Colored networks can again be decomposed along the connectivity structure. For rooted 3connected cubic planar graphs embedded in the plane, we switch to the dual and count rooted triangulations. Since all these numbers can be evaluated in polynomial time using dynamic programming, rooted connected cubic planar graphs can be generated uniformly at random in polynomial time by inverting the decomposition along the connectivity structure. To generate connected cubic planar graphs without a root uniformly at random, we apply rejection sampling and obtain an expected polynomial time algorithm.