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31
Optimal Graph Orientation with Storage Applications
, 1995
"... We show that the edges of a graph with maximum edge density d can always be oriented such that each vertex has indegree at most d. Hence, for arbitrary graphs, edges can always be assigned to incident vertices as uniformly as possible. For example, indegree 3 is achieved for planar graphs. This im ..."
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We show that the edges of a graph with maximum edge density d can always be oriented such that each vertex has indegree at most d. Hence, for arbitrary graphs, edges can always be assigned to incident vertices as uniformly as possible. For example, indegree 3 is achieved for planar graphs. This immediately gives a spaceoptimal data structure that answers edge membership queries in a maximum edge densityd graph in O(log d) time. Keywords Graph orientation, edge density, Hall condition, balanced adjacency lists, edge membership queries 1 The Theorem Let G be an undirected graph with n vertices and m edges. The parameter ffi(G) = m n is commonly called the edge density of G. The maximum (edge) density is the smallest integer d such that the edge density of any subgraph of G does not exceed d. More precisely, d = dmaxfffi(G 0 ) j G 0 is a subgraph of Gge. For example, d 1 for trees, d 3 for planar graphs, d = d 1 2 log 2 ne for hypercubes [GG,AH], and d d 1 2 (c \Gamma...
Planar Graphs with Topological Constraints
 Journal of Graph Algorithms and Applications
, 2002
"... We address in this paper the problem of constructing embeddings of planar graphs satisfying declarative, userdefined topological constraints. The constraints consist each of a cycle... ..."
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We address in this paper the problem of constructing embeddings of planar graphs satisfying declarative, userdefined topological constraints. The constraints consist each of a cycle...
A linear time algorithm for finding a maximal planar subgraph based on PCtrees
 of Lecture Notes in Computer Science
, 2005
"... ABSTRACT. Given an undirected graph G, the maximal planar subgraph problem is to determine a planar subgraph H of G such that no edge of GH can be added to H without destroying planarity. Polynomial algorithms have been obtained by Jakayumar, Thulasiraman and Swamy [6] and Wu [9]. O(mlogn) algorith ..."
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ABSTRACT. Given an undirected graph G, the maximal planar subgraph problem is to determine a planar subgraph H of G such that no edge of GH can be added to H without destroying planarity. Polynomial algorithms have been obtained by Jakayumar, Thulasiraman and Swamy [6] and Wu [9]. O(mlogn) algorithms were previously given by Di Battista and Tamassia [3] and Cai, Han and Tarjan [2]. A recent O(mα (n)) algorithm was obtained by La Poute [7]. Our algorithm is based on a simple planarity test [5] developed by the author, which is a vertex addition algorithm based on a depthfirstsearch ordering. The planarity test [5] uses no complicated data structure and is conceptually simpler than Hopcroft and Tarjan's path addition and Lempel, Even and Cederbaum's vertex addition approaches. 1 1.
An Implementation of the Hopcroft and Tarjan Planarity Test and Embedding Algorithm
, 1993
"... We describe an implementation of the Hopcroft and Tarjan planarity test and embedding algorithm. The program tests the planarity of the input graph and either constructs a combinatorial embedding (if the graph is planar) or exhibits a Kuratowski subgraph (if the graph is nonplanar). ..."
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We describe an implementation of the Hopcroft and Tarjan planarity test and embedding algorithm. The program tests the planarity of the input graph and either constructs a combinatorial embedding (if the graph is planar) or exhibits a Kuratowski subgraph (if the graph is nonplanar).
An efficient implementation of the PCtrees algorithm of shih and hsu’s planarity test
 Institute of Information Science, Academia Sinica
, 2003
"... In Shih & Hsu [9] a simpler planarity test was introduced utilizing a data structure called PCtrees (generalized from PQtrees). In this paper we give an efficient implementation of that linear time algorithm and illustrate in detail how to obtain a Kuratowski subgraph when the given graph is not p ..."
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In Shih & Hsu [9] a simpler planarity test was introduced utilizing a data structure called PCtrees (generalized from PQtrees). In this paper we give an efficient implementation of that linear time algorithm and illustrate in detail how to obtain a Kuratowski subgraph when the given graph is not planar, and how to obtain the embedding alongside the testing algorithm. We have implemented the algorithm using LEDA and an object code is available at
I/OEfficient Planar Separators and Applications
, 2001
"... We present a new algorithm to compute a subset S of vertices of a planar graph G whose removal partitions G into O(N/h) subgraphs of size O(h) and with boundary size O( p h) each. The size of S is O(N= p h). Computing S takes O(sort(N)) I/Os and linear space, provided that M 56hlog² B. Together with ..."
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We present a new algorithm to compute a subset S of vertices of a planar graph G whose removal partitions G into O(N/h) subgraphs of size O(h) and with boundary size O( p h) each. The size of S is O(N= p h). Computing S takes O(sort(N)) I/Os and linear space, provided that M 56hlog² B. Together with recent reducibility results, this leads to O(sort(N)) I/O algorithms for breadthfirst search (BFS), depthfirst search (DFS), and single source shortest paths (SSSP) on undirected embedded planar graphs. Our separator algorithm does not need a BFS tree or an embedding of G to be given as part of the input. Instead we argue that "local embeddings" of subgraphs of G are enough.
Drawing Database Schemas
 Software  Practice and Experience
"... A wide number of practical applications would benefit from automatically generated graphical representations of database schemas, in which tables are represented by boxes, and table attributes correspond to distinct stripes inside each table. Links, connecting attributes of two different tables, rep ..."
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A wide number of practical applications would benefit from automatically generated graphical representations of database schemas, in which tables are represented by boxes, and table attributes correspond to distinct stripes inside each table. Links, connecting attributes of two different tables, represent referential constraints or join relationships, and may attach arbitrarily to the left or to the right side of the stripes representing the attributes. To our knowledge no drawing technique is available to automatically produce diagrams in such strongly constrained drawing convention. In this paper we provide a polynomial time algorithm for solving this problem and test its efficiency and effectiveness against a large test suite. Also, we describe an implementation of a system that uses such an algorithm and we study the main methodological problems we faced in developing such a technology.
I/OOptimal Planar Embedding Using Graph Separators
, 2001
"... We present a new algorithm to test whether a given graph G is planar and to compute a planar embedding G of G if such an embedding exists. Our algorithm utilizes a fundamentally new approach based on graph separators to obtain such an embedding. The I/Ocomplexity of our algorithm is O(sort(N)). A s ..."
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We present a new algorithm to test whether a given graph G is planar and to compute a planar embedding G of G if such an embedding exists. Our algorithm utilizes a fundamentally new approach based on graph separators to obtain such an embedding. The I/Ocomplexity of our algorithm is O(sort(N)). A simple simulation technique reduces the I/Ocomplexity of our algorithm to O(perm(N)). We prove a matching lower bound of W(perm(N)) I/Os for computing a planar embedding of a given planar graph.
Terrain Guarding is NPHard
, 2009
"... A set G of points on a 1.5dimensional terrain, also known as an xmonotone polygonal chain, is said to guard the terrain if every point on the terrain is seen by a point in G. Two points on the terrain see each other if and only if the line segment between them is never strictly below the terrain. ..."
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A set G of points on a 1.5dimensional terrain, also known as an xmonotone polygonal chain, is said to guard the terrain if every point on the terrain is seen by a point in G. Two points on the terrain see each other if and only if the line segment between them is never strictly below the terrain. The minimum terrain guarding problem asks for a minimum guarding set for the given input terrain. Using a reduction from PLANAR 3SAT we prove that the decision version of this problem is NPhard. This solves a significant open problem and complements recent positive approximability results for the optimization problem. 1