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Compact Encodings of Planar Graphs via Canonical Orderings and Multiple Parentheses
, 1998
"... . We consider the problem of coding planar graphs by binary strings. Depending on whether O(1)time queries for adjacency and degree are supported, we present three sets of coding schemes which all take linear time for encoding and decoding. The encoding lengths are significantly shorter than th ..."
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. We consider the problem of coding planar graphs by binary strings. Depending on whether O(1)time queries for adjacency and degree are supported, we present three sets of coding schemes which all take linear time for encoding and decoding. The encoding lengths are significantly shorter than the previously known results in each case. 1 Introduction This paper investigates the problem of encoding a graph G with n nodes and m edges into a binary string S. This problem has been extensively studied with three objectives: (1) minimizing the length of S, (2) minimizing the time needed to compute and decode S, and (3) supporting queries efficiently. A number of coding schemes with different tradeoffs have been proposed. The adjacencylist encoding of a graph is widely useful but requires 2mdlog ne bits. (All logarithms are of base 2.) A folklore scheme uses 2n bits to encode a rooted nnode tree into a string of n pairs of balanced parentheses. Since the total number of such trees is...
Graph Drawing '93
, 1993
"... not Available. Characterizing Proximity Trees Prosenjit Bose, William Lenhart, y and Giuseppe Liotta z Much attention has been given over the past several years to developing algorithms for embedding abstract graphs in the plane such that the resulting drawing has certain geometric properties ..."
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Cited by 3 (3 self)
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not Available. Characterizing Proximity Trees Prosenjit Bose, William Lenhart, y and Giuseppe Liotta z Much attention has been given over the past several years to developing algorithms for embedding abstract graphs in the plane such that the resulting drawing has certain geometric properties. For example, those graphs which admit planar drawings have been completely characterized and efficient algorithms for producing planar drawings of these graphs have been designed ([4], [9]). For an overview of graph drawing problems and algorithms, the reader is referred to the excellent bibliography of Di Battista, Eades, Tamassia and Tollis [2]. Moreover, many problems in pattern recognition and classification, geographic variation analysis, geographic information systems, computational geometry, computational morphology, and computer vision use the underlying structure present in a set of data points revealed by means of a proximity graph. A proximity graph attempts to exhibit the rela...
Parallel Computation and Graphical Visualization of the Minimum Crossing Number of a Graph
, 1998
"... Finding the minimum crossing number of a graph is an interesting and challenging problem in graph theory and applied mathematics. Real world applications of this problem, such as circuit layout and network design, are becoming more and more important. This thesis presents a parallel algorithm for f ..."
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Finding the minimum crossing number of a graph is an interesting and challenging problem in graph theory and applied mathematics. Real world applications of this problem, such as circuit layout and network design, are becoming more and more important. This thesis presents a parallel algorithm for finding the minimum crossing number of a graph, based on the first sequential algorithm presented in [20]. This parallel algorithm was tested on various architectures and a comparison of the corresponding results is given, including running time, efficiency, and speedup. Implementation of the algorithm gives us an ability to verify conjectures proposed for various families of graphs, as well as apply the algorithm to real world applications. Another important aspect of the problem is ability to draw the solution on the 2 \Gamma D plane. This thesis gives an overview of graph drawing algorithms, starting with the famous result by F'ary, and presents a new algorithm for drawing complete graphs...
A Simple Linear Time Algorithm for Embedding Maximal Planar Graphs
, 1993
"... All existing algorithms for planarity testing/planar embedding can be grouped into two principal classes. Either, they run in linear time, but to the expense of complex algorithmic concepts or complex datastructures, or they are easy to understand and implement, but require more than linear time [ ..."
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Cited by 2 (1 self)
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All existing algorithms for planarity testing/planar embedding can be grouped into two principal classes. Either, they run in linear time, but to the expense of complex algorithmic concepts or complex datastructures, or they are easy to understand and implement, but require more than linear time [2]. In this paper, a new lineartime algorithm for embedding maximal planar graphs is proposed. This algorithm is both easy to understand and easy to implement. The algorithm consists of three phases which use only simple, local graphmodifications. In addition to planar embedding, the new algorithm allows to test graphs for maximal planarity. The generation of Straight Line Drawings by a technique of Read [12] can be naturally incorporated into the algorithm. We also demonstrate how to generate random (maximal) planar graphs. The algorithm presented constitutes a first step towards a simple, lineartime solution for embedding general planar graphs. 2 3 1 Introduction One of the main top...
GUItar and FAgoo: Graphical interface for automata visualization, editing, and interaction ⋆
"... Abstract. GUItar is a graphical environment for graph visualization, editing, and interaction, that specially focuses in finite automata diagrams. The application incorporates mechanisms to facilitate the editing of these graphs. It also provides a style manager that allows the creation of rich stat ..."
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Abstract. GUItar is a graphical environment for graph visualization, editing, and interaction, that specially focuses in finite automata diagrams. The application incorporates mechanisms to facilitate the editing of these graphs. It also provides a style manager that allows the creation of rich state and arc styles to be used in the drawing of its objects. This style manager allows the system to cope with complex styles, broaden the application scope to graphical representations of other computational models like transducers or Turing machines. GUItar also has a foreign function call (FFC) mechanism for the easy integration of external modules and libraries like automata symbolic manipulators or graph drawing libraries. For automatic graph drawing we are developing FAgoo, a package that seeks to provide tools capable of finding pleasant graph drawings. FAgoo implements graph drawing algorithms that find embeddings which the user, with minimal manual changes, can adjust to its aesthetically taste. Both GUItar and FAgoo are on going projects licensed under GPL. 1
Grid Embedding of Internally Triangulated Plane Graphs without Nonempty Triangles
, 1995
"... A straight line grid embedding of a plane graph G is a drawing of G such that the vertices are drawn at grid points and the edges are drawn as nonintersecting straight line segments. In this paper, we show that, if an internally triangulated plane graph G has no nonempty triangles (a nonempty tri ..."
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A straight line grid embedding of a plane graph G is a drawing of G such that the vertices are drawn at grid points and the edges are drawn as nonintersecting straight line segments. In this paper, we show that, if an internally triangulated plane graph G has no nonempty triangles (a nonempty triangle is a triangle of G containing some vertices in its interior), then G can be embedded on a grid of size W \Theta H such that W + H n, W (n + 3)=2 and H 2(n \Gamma 1)=3, where n is the number of vertices of G. Such an embedding can be computed in linear time. 1 Introduction Let G = (V; E) be a graph with n vertices. We always assume n 3 in this paper. G is planar if it can be drawn on the plane such that the vertices are located at distinct points, and the edges are represented by nonintersecting curves joining their endpoints. A plane graph is a planar graph with a fixed plane embedding. A straight line grid embedding of a planar graph is a drawing where the vertices are located at...
Straightline Drawings of 1planar Graphs
"... Abstract. The classical Fáry’s theorem from the 1930s states that every planar graph can be drawn as a straightline drawing. In this paper, we extend Fáry’s theorem to nonplanar graphs. More specifically, we study the problem of drawing 1planar graphs with straightline edges. A 1planar graph is ..."
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Abstract. The classical Fáry’s theorem from the 1930s states that every planar graph can be drawn as a straightline drawing. In this paper, we extend Fáry’s theorem to nonplanar graphs. More specifically, we study the problem of drawing 1planar graphs with straightline edges. A 1planar graph is a sparse nonplanar graph with at most one crossing per edge. We give a characterisation of those 1planar graphs that admit a straightline drawing. The proof of the characterisation consists of a linear time testing algorithm and a drawing algorithm. We also show that there are 1planar graphs for which every straightline drawing has exponential area. To our best knowledge, this is the first result to extend Fáry’s theorem to nonplanar graphs. 1
Compact Encodings of Planar Graphs via Canonical Orderings and Multiple Parentheses ∗
, 2008
"... Let G be a plane graph of n nodes, m edges, f faces, and no selfloop. G need not be connected or simple (i.e., free of multiple edges). We give three sets of coding schemes for G which all take O(m + n) time for encoding and decoding. Our schemes employ new properties of canonical orderings for pla ..."
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Let G be a plane graph of n nodes, m edges, f faces, and no selfloop. G need not be connected or simple (i.e., free of multiple edges). We give three sets of coding schemes for G which all take O(m + n) time for encoding and decoding. Our schemes employ new properties of canonical orderings for planar graphs and new techniques of processing strings of multiple types of parentheses. For applications that need to determine in O(1) time the adjacency of two nodes and the degree of a node, we use 2m + (5 + 1 k)n + o(m + n) bits for any constant k> 0 while the best previous bound by Munro and Raman is 2m + 8n + o(m + n). If G is triconnected or triangulated, our bit count decreases to 2m + 3n + o(m + n) or 2m + 2n + o(m + n), respectively. If G is simple, our bit count is 5 1 3m + (5 + k)n + o(n) for any constant k> 0. Thus, if a simple G is also triconnected or triangulated, then 2m + 2n + o(n) or 2m + n + o(n) bits suffice, respectively. If only adjacency queries are supported, the bit counts for a general G and a simple G become 2m + 14 4 3 n + o(m + n) and 3m + 5n + o(n), respectively. If we only need to reconstruct G from its code, a simple and triconnected G uses
Universal Point Subsets . . .
"... A set S of k points in the plane is a universal point subset for a class G of planar graphs if every graph belonging to G admits a planar straightline drawing such that k of its vertices are represented by the points of S. In this paper we study the following main problem: For a given class of gra ..."
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A set S of k points in the plane is a universal point subset for a class G of planar graphs if every graph belonging to G admits a planar straightline drawing such that k of its vertices are represented by the points of S. In this paper we study the following main problem: For a given class of graphs, what is the maximum k such that there exists a universal point subset of size k? We provide a ⌈ √ n ⌉ lower bound on k for the class of planar graphs with n vertices. In addition, we consider the value F (n, G) such that every set of F (n, G) points in general position is a universal subset for all graphs with n vertices belonging to the family G, and we establish upper and lower bounds for F (n, G) for different families of planar graphs, including 4connected planar graphs and nestedtriangles graphs.