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Inserting an Edge Into a Planar Graph
 Algorithmica
, 2000
"... Computing a crossing minimum drawing of a given planar graph G augmented by an additional edge e in which all crossings involve e, has been a long standing open problem in graph drawing. Alternatively, the problem can be stated as finding a planar combinatorial embedding of a planar graph G in which ..."
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Cited by 18 (9 self)
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Computing a crossing minimum drawing of a given planar graph G augmented by an additional edge e in which all crossings involve e, has been a long standing open problem in graph drawing. Alternatively, the problem can be stated as finding a planar combinatorial embedding of a planar graph G in which the given edge e can be inserted with the minimum number of crossings. Many problems concerned with the optimization over the set of all combinatorial embeddings of a planar graph turned out to be NPhard. Surprisingly, we found a conceptually simple linear time algorithm based on SPQRtrees, which is able to find a crossing minimum solution.
Fast Incremental Planarity Testing
 19 th International Colloquium on Automata, Languages and Programming (ICALP), volume 623 of LNCS
, 1992
"... The incremental planarity testing problem is to perform the following operations on a biconnected planar graph G of at most n vertices: test if an edge can be added between two vertices while preserving planarity; add edges and vertices that preserve planarity. Let m be the total number of operation ..."
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Cited by 16 (0 self)
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The incremental planarity testing problem is to perform the following operations on a biconnected planar graph G of at most n vertices: test if an edge can be added between two vertices while preserving planarity; add edges and vertices that preserve planarity. Let m be the total number of operations. We present fast data structures for this problem that can be used in conjunction with the previous algorithm of Di Battista and Tamassia to achieve an O(ff(m; n)) worstcase amortized time per test operation. If the graph is biconnected, a sequence of n additions can be performed in total time O(mff(m;n)) worstcase plus O(n) expected time. Our tree data structure is flexible and can answer in O(1) time queries about parents, roots, and nearest common ancestors while performing tree modifications such as inserting nodes, cutting edges, and merging or splitting nodes. If the graph is not biconnected then insertions of edges and vertices require O(log n) amortized expected time per operat...
Graph Drawing
 Lecture Notes in Computer Science
, 1997
"... INTRODUCTION Graph drawing addresses the problem of constructing geometric representations of graphs, and has important applications to key computer technologies such as software engineering, database systems, visual interfaces, and computeraideddesign. Research on graph drawing has been conducte ..."
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Cited by 14 (3 self)
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INTRODUCTION Graph drawing addresses the problem of constructing geometric representations of graphs, and has important applications to key computer technologies such as software engineering, database systems, visual interfaces, and computeraideddesign. Research on graph drawing has been conducted within several diverse areas, including discrete mathematics (topological graph theory, geometric graph theory, order theory), algorithmics (graph algorithms, data structures, computational geometry, vlsi), and humancomputer interaction (visual languages, graphical user interfaces, software visualization). This chapter overviews aspects of graph drawing that are especially relevant to computational geometry. Basic definitions on drawings and their properties are given in Section 1.1. Bounds on geometric and topological properties of drawings (e.g., area and crossings) are presented in Section 1.2. Section 1.3 deals with the time complexity of fundamental graph drawin
Maintaining information in fullydynamic trees with top trees
 ACM Transactions on Algorithms
, 2003
"... We introduce top trees as a design of a new simpler interface for data structures maintaining information in a fullydynamic forest. We demonstrate how easy and versatile they are to use on a host of different applications. For example, we show how to maintain the diameter, center, and median of eac ..."
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Cited by 12 (0 self)
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We introduce top trees as a design of a new simpler interface for data structures maintaining information in a fullydynamic forest. We demonstrate how easy and versatile they are to use on a host of different applications. For example, we show how to maintain the diameter, center, and median of each tree in the forest. The forest can be updated by insertion and deletion of edges and by changes to vertex and edge weights. Each update is supported in O(log n) time, where n is the size of the tree(s) involved in the update. Also, we show how to support nearest common ancestor queries and level ancestor queries with respect to arbitrary roots in O(log n) time. Finally, with marked and unmarked vertices, we show how to compute distances to a nearest marked vertex. The later has applications to approximate nearest marked vertex in general graphs, and thereby to static optimization problems over shortest path metrics. Technically speaking, top trees are easily implemented either with Frederickson’s topology trees [Ambivalent Data Structures for Dynamic 2EdgeConnectivity and k Smallest Spanning Trees, SIAM J. Comput. 26 (2) pp. 484–538, 1997] or with Sleator and Tarjan’s dynamic
OutputSensitive Reporting of Disjoint Paths
, 1996
"... A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For ..."
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Cited by 11 (2 self)
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A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For k < 3, we present an optimal data structure for G that uses O(n) space and executes kpath queries in outputsensitive O() time. For triconnected planar graphs, our results make use of a new combinatorial structure that plays the same role as bipolar (st) orientations for biconnected planar graphs. This combinatorial structure also yields an alternative construction of convex grid drawings of triconnected planar graphs.
Dynamic Expression Trees
, 1991
"... We present a technique for dynamically maintaining a collection of arithmetic expressions represented by binary trees (whose leaves are variables and whose internal nodes are operators). A query operation asks for the value of an expression (associated with the root of a tree). Update operations inc ..."
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Cited by 11 (3 self)
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We present a technique for dynamically maintaining a collection of arithmetic expressions represented by binary trees (whose leaves are variables and whose internal nodes are operators). A query operation asks for the value of an expression (associated with the root of a tree). Update operations include changing the value of a variable and combining or decomposing expressions by linking or cutting the corresponding trees. Our dynamic data structure uses linear space and supports queries and updates in logarithmic time. An important application is the dynamic maintenance of maximum flow and shortest path in seriesparallel digraphs under a sequence of vertex and edge insertions, series and parallel compositions, and their respective inverses. Queries include reporting the maximum flow or shortest stpath in a seriesparallel subgraph.
Drawing Directed Acyclic Graphs: An Experimental Study
, 1996
"... In this paper we consider the class of directed acyclic graphs (DAGs), and present the results of an experimental study on four drawing algorithms specifically developed for DAGs. Our study is conducted on two large test suites of DAGs and yields more than 30 charts comparing the performance of ..."
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Cited by 11 (0 self)
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In this paper we consider the class of directed acyclic graphs (DAGs), and present the results of an experimental study on four drawing algorithms specifically developed for DAGs. Our study is conducted on two large test suites of DAGs and yields more than 30 charts comparing the performance of the drawing algorithms with respect to several quality measures, including area, crossings, bends, and aspect ratio. The algorithms exhibit various tradeoffs with respect to the quality measures, and none of them clearly outperforms the others.
Algorithms for Drawing Media
"... We describe algorithms for drawing media, systems of states, tokens and actions that have state transition graphs in the form of partial cubes. Our algorithms are based on two principles: embedding the state transition graph in a lowdimensional integer lattice and projecting the lattice onto the p ..."
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Cited by 10 (4 self)
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We describe algorithms for drawing media, systems of states, tokens and actions that have state transition graphs in the form of partial cubes. Our algorithms are based on two principles: embedding the state transition graph in a lowdimensional integer lattice and projecting the lattice onto the plane, or drawing the medium as a planar graph with centrally symmetric faces.
Graph Planarization and Skewness
"... The problem of finding a maximum spanning planar subgraph of a nonplanar graph is NPComplete. Several heuristics for the problem have been devised but their worstcase performance is unknown, although a trivial lower bound of 1/3 the optimum number of edges is easily shown. We discuss a new heurist ..."
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Cited by 10 (0 self)
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The problem of finding a maximum spanning planar subgraph of a nonplanar graph is NPComplete. Several heuristics for the problem have been devised but their worstcase performance is unknown, although a trivial lower bound of 1/3 the optimum number of edges is easily shown. We discuss a new heuristic, based on spanning trees, for generating a subgraph with size at least 2/3 of the optimum for any input graph. The skewness of the ndimensional hypercube Qn is also derived. Finally, we explore the relationship between the skewness and crossing number of a graph.