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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.
An InformationTheoretic Upper Bound on Planar Graphs Using WellOrderly Maps
, 2011
"... This chapter deals with compressed coding of graphs. We focus on planar graphs, a widely studied class of graphs. A planar graph is a graph that admits an embedding in the plane without edge crossings. Planar maps (class of embeddings of a planar graph) are easier to study than planar graphs, but a ..."
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Cited by 26 (3 self)
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This chapter deals with compressed coding of graphs. We focus on planar graphs, a widely studied class of graphs. A planar graph is a graph that admits an embedding in the plane without edge crossings. Planar maps (class of embeddings of a planar graph) are easier to study than planar graphs, but as a planar graph may admit an exponential number of maps, they give little information on graphs. In order to give an informationtheoretic upper bound on planar graphs, we introduce a definition of a quasicanonical embedding for planar graphs: wellorderly maps. This appears to be an useful tool to study and encode planar graphs. We present upper bounds on the number of unlabeled planar graphs and on the number of edges in a random planar graph. We also present an algorithm to compute wellorderly maps and implying an efficient coding of planar graphs.
Canonical Decomposition of Outerplanar Maps and Application to Enumeration, Coding and Generation
, 2003
"... In this article we define a canonical decomposition of rooted outerplanar maps into a spanning tree and a list of edges. This decomposition, constructible in linear time, implies the existence of bijection between rooted outerplanar maps with n nodes and bicolored rooted ordered trees with n node ..."
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Cited by 14 (1 self)
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In this article we define a canonical decomposition of rooted outerplanar maps into a spanning tree and a list of edges. This decomposition, constructible in linear time, implies the existence of bijection between rooted outerplanar maps with n nodes and bicolored rooted ordered trees with n nodes where all the nodes of the last branch are colored white. As a consequence, for rooted outerplanar maps of n nodes, we derive: an enumeration formula, and an asymptotic of 2 3n (log n) ; an optimal data structure of asymptotically 3n bits, built in O(n) time, supporting adjacency and degree queries in worstcase constant time and neighbors query of a ddegree node in worstcase O(d) time...
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.
Composable Art: Objects that can be arranged in many ways
"... Abstract The concept of composable art is introduced and four examples of composable art objects are given. We analyze the number of different compositions that can be made for each object using combinatorics and the dynamic programming technique. ..."
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Abstract The concept of composable art is introduced and four examples of composable art objects are given. We analyze the number of different compositions that can be made for each object using combinatorics and the dynamic programming technique.