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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 12 (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 Generation of Dags for Graph Drawing
, 2000
"... The Graph Drawing community uses test suites for comparing layout quality and efficiency. Those suites often claim to collect randomly generated graphs, although in most cases randomness is a loosely defined notion. Wepropose a simple algorithm for generating acyclic digraphs with a given number of ..."
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Cited by 11 (1 self)
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The Graph Drawing community uses test suites for comparing layout quality and efficiency. Those suites often claim to collect randomly generated graphs, although in most cases randomness is a loosely defined notion. Wepropose a simple algorithm for generating acyclic digraphs with a given number of vertices uniformly at random. Applying standard combinatorial techniques, we describe the overall shape and average edge density of an acyclic digraph. The usefulness of our algorithm resides in the possibility of controlling edge density of the generated graphs. We have used our technique to build a large test suite of acyclic digraphs with various edge density and number of vertices ranging from 10 to 1000.
Generating outerplanar graphs uniformly at random
 in Combinatorics, Probability, and Computation
, 2003
"... supported by the DFG (GRK 588/1) Abstract. We show how to generate labeled and unlabeled outerplanar graphs with n vertices uniformly at random in polynomial time in n. To generate labeled outerplanar graphs, we present a counting technique using the decomposition of a graph according to its block s ..."
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Cited by 9 (6 self)
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supported by the DFG (GRK 588/1) Abstract. We show how to generate labeled and unlabeled outerplanar graphs with n vertices uniformly at random in polynomial time in n. To generate labeled outerplanar graphs, we present a counting technique using the decomposition of a graph according to its block structure, and compute the exact number of labeled outerplanar graphs. This allows us to make the correct probabilistic choices in a recursive generation of uniformly distributed outerplanar graphs. Next we modify our formulas to count rooted unlabeled graphs, and finally show how to use these formulas in a Las Vegas algorithm to generate unlabeled outerplanar graphs uniformly at random in expected polynomial time. random structures, outerplanar graphs, efficient counting, uniform generation 1
Computing A DiameterConstrained Minimum Spanning Tree
, 2001
"... In numerous practical applications, it is necessary to find the smallest possible tree with a bounded diameter. A diameterconstrained minimum spanning tree (DCMST) of a given undirected, edgeweighted graph, G, is the smallestweight spanning tree of all spanning trees of G which contain no path wi ..."
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Cited by 8 (0 self)
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In numerous practical applications, it is necessary to find the smallest possible tree with a bounded diameter. A diameterconstrained minimum spanning tree (DCMST) of a given undirected, edgeweighted graph, G, is the smallestweight spanning tree of all spanning trees of G which contain no path with more than k edges, where k is a given positive integer. The problem of finding a DCMST is NPcomplete for all values of k; 4 k (n  2), except when all edgeweights are identical. A DCMST is essential for the efficiency of various distributed mutual exclusion algorithms, where it can minimize the number of messages communicated among processors per critical section. It is also useful in linear lightwave networks, where it can minimize interference in the network by limiting the traffic in the network lines. Another practical application requiring a DCMST arises in data compression, where some algorithms compress a file utilizing a tree datastructure, and decompress a path in the tree to access a record. A DCMST helps such algorithms to be fast without sacrificing a lot of storage space. We present a survey of the literature on the DCMST problem, study the expected diameter of a random labeled tree, and present five new polynomialtime algorithms for an approximate DCMST. One of our new algorithms constructs an approximate DCMST in a modified greedy fashion, employing a heuristic for selecting an edge to be added to iii the tree in each stage of the construction. Three other new algorithms start with an unconstrained minimum spanning tree, and iteratively refine it into an approximate DCMST. We also present an algorithm designed for the special case when the diameter is required to be no more than 4. Such a diameter4 tree is also used for evaluating the quality of o...
Chapter 2 An InformationTheoretic Upper Bound on Planar Graphs Using WellOrderly Maps
"... Abstract 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 graph ..."
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Abstract 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 1 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.