Results 1  10
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24
Radial Level Planarity Testing and Embedding in Linear Time
 Journal of Graph Algorithms and Applications
, 2005
"... A graph with a given partition of the vertices on k concentric circles is radial level planar if there is a vertex permutation such that the edges can be routed strictly outwards without crossings. Radial level planarity extends level planarity, where the vertices are placed on k horizontal lines an ..."
Abstract

Cited by 19 (9 self)
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A graph with a given partition of the vertices on k concentric circles is radial level planar if there is a vertex permutation such that the edges can be routed strictly outwards without crossings. Radial level planarity extends level planarity, where the vertices are placed on k horizontal lines and the edges are routed strictly downwards without crossings. The extension is characterised by rings, which are level nonplanar biconnected components. Our main results are linear time algorithms for radial level planarity testing and for computing an embedding. We introduce PQRtrees as a new data structure where Rnodes and associated templates for their manipulation are introduced to deal with rings. Our algorithms extend level planarity testing and embedding algorithms which use PQtrees.
Fully Dynamic Algorithm for Recognition and Modular Decomposition of Permutation Graphs
 ALGORITHMICA
, 2009
"... This paper considers the problem of maintaining a compact representation (O(n) space) of permutation graphs under vertex and edge modifications (insertion or deletion). That representation allows us to answer adjacency queries in O(1) time. The approach is based on a fully dynamic modular decomposit ..."
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Cited by 7 (4 self)
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This paper considers the problem of maintaining a compact representation (O(n) space) of permutation graphs under vertex and edge modifications (insertion or deletion). That representation allows us to answer adjacency queries in O(1) time. The approach is based on a fully dynamic modular decomposition algorithm for permutation graphs that works in O(n) time per edge and vertex modification. We thereby obtain a fully dynamic algorithm for the recognition of permutation graphs.
Inserting a vertex into a planar graph
 In ACMSIAM Symposium on Discrete Algorithms 2009; ACM Press
, 2009
"... We consider the problem of computing a crossing minimum drawing of a given planar graph G = (V, E) augmented by a star, i.e., an additional vertex v together with its incident edges Ev = {(v, u)  u ∈ V}, in which all crossings involve Ev. Alternatively, the problem can be stated as finding a plana ..."
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Cited by 6 (6 self)
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We consider the problem of computing a crossing minimum drawing of a given planar graph G = (V, E) augmented by a star, i.e., an additional vertex v together with its incident edges Ev = {(v, u)  u ∈ V}, in which all crossings involve Ev. Alternatively, the problem can be stated as finding a planar embedding of G, in which the given star can be inserted requiring the minimum number of crossings. This is a generalization of the crossing minimum edge insertion problem [15], and can help to find improved approximations for the crossing minimization problem. Indeed, in practice, the algorithm for the crossing minimum edge insertion problem turned out to be the key for obtaining the currently strongest approximate solutions for the crossing number of general graphs. The generalization considered here can lead to even better solutions for the crossing minimization problem. Furthermore, it offers new insight into the crossing number problem for almostplanar and apex graphs. It has been an open problem whether the star insertion problem is polynomially solvable. We give an affirmative answer by describing the first efficient algorithm for this problem. This algorithm uses the SPQRtree data structure to handle the exponential number of possible embeddings, in conjunction with dynamic programming schemes for which we introduce partitioning cost subproblems. 1
Testing Planarity of Partially Embedded Graphs
, 2009
"... We study the following problem: Given a planar graph G and a planar drawing (embedding) of a subgraph of G, can such a drawing be extended to a planar drawing of the entire graph G? This problem fits the paradigm of extending a partial solution to a complete one, which has been studied before in man ..."
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Cited by 5 (2 self)
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We study the following problem: Given a planar graph G and a planar drawing (embedding) of a subgraph of G, can such a drawing be extended to a planar drawing of the entire graph G? This problem fits the paradigm of extending a partial solution to a complete one, which has been studied before in many different settings. Unlike many cases, in which the presence of a partial solution in the input makes hard an otherwise easy problem, we show that the planarity question remains polynomialtime solvable. Our algorithm is based on several combinatorial lemmata which show that the planarity of partially embedded graphs meets the “oncas” behaviour – obvious necessary conditions for planarity are also sufficient. These conditions are expressed in terms of the interplay between (a) rotation schemes and containment relationships between cycles and (b) the decomposition of a graph into its connected, biconnected, and triconnected components. This implies that no dynamic programming is needed for a decision algorithm and that the elements of the decomposition can be processed independently. Further, by equipping the components of the decomposition with suitable data structures and by carefully splitting the problem into simpler subproblems, we improve our algorithm to reach lineartime complexity. Finally, we consider several generalizations of the problem, e.g. minimizing the number of edges of the partial embedding that need to be rerouted to extend it, and argue that they are NPhard. Also, we show how our algorithm can be applied to solve related Graph Drawing problems.
Optimal Labeling for Connectivity Checking in Planar Networks with Obstacles
, 2009
"... We consider the problem of determining in a planar graph G whether two vertices x and y are linked by a path that avoids a set X of vertices and a set F of edges. We attach labels to vertices in such a way that this fact can be determined from the labels of x and y, the vertices in X and the ends of ..."
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Cited by 4 (3 self)
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We consider the problem of determining in a planar graph G whether two vertices x and y are linked by a path that avoids a set X of vertices and a set F of edges. We attach labels to vertices in such a way that this fact can be determined from the labels of x and y, the vertices in X and the ends of the edges of F. For a planar graph with n vertices, we construct labels of size O(log n). The problem is motivated by the need to quickly compute alternative routes in networks under node or edge failures.
Twopage book embedding and clustered graph planarity
, 2009
"... Abstract: A 2page book embedding of a graph places the vertices linearly on a spine (a line segment) and the edges on two pages (two half planes sharing the spine) so that each edge is embedded in one of the pages without edge crossings. Testing whether a given graph admits a 2page book embedding ..."
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Cited by 3 (0 self)
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Abstract: A 2page book embedding of a graph places the vertices linearly on a spine (a line segment) and the edges on two pages (two half planes sharing the spine) so that each edge is embedded in one of the pages without edge crossings. Testing whether a given graph admits a 2page book embedding is known to be NPcomplete. In this paper, we study the problem of testing whether a given graph admits a 2page book embedding with a fixed edge partition. Based on structural properties of planar graphs, we prove that the problem of testing and finding a 2page book embedding of a graph with a partitioned edge set can be solved in linear time. As an application of our main result, we consider the problem of testing planarity of clustered graphs. The complexity of testing clustered graph planarity is a long standing open problem in Graph Drawing. Recently, polynomial time algorithms have been found for several classes of clustered graphs in which both the structure of the underlying graphs and clustering structure are restricted. However, when the underlying graph is disconnected, the problem remains open. Our book embedding results imply that the clustered planarity problem can be solved in linear time for a certain class of clustered graphs with arbitrary underlying graphs (i.e. possibly disconnected). 1
Testing Simultaneous Planarity when the Common Graph is 2Connected
, 2011
"... Two planar graphs G1 and G2 sharing some vertices and edges are simultaneously planar if they have planar drawings such that a shared vertex [edge] is represented by the same point [curve] in both drawings. It is an open problem whether simultaneous planarity can be tested efficiently. We give a lin ..."
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Cited by 3 (1 self)
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Two planar graphs G1 and G2 sharing some vertices and edges are simultaneously planar if they have planar drawings such that a shared vertex [edge] is represented by the same point [curve] in both drawings. It is an open problem whether simultaneous planarity can be tested efficiently. We give a lineartime algorithm to test simultaneous planarity when the two graphs share a 2connected subgraph. Our algorithm extends to the case of k planar graphs where each vertex [edge] is either common to all graphs or belongs to exactly one of them, and the common subgraph is 2connected.
Fáry’s Theorem for 1Planar Graphs
"... Abstract. Fáry’s theorem states that every plane graph can be drawn as a straightline drawing. A plane graph is a graph embedded in a plane without edge crossings. In this paper, we extend Fáry’s theorem to nonplanar graphs. More specifically, we study the problem of drawing 1plane graphs with str ..."
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Cited by 2 (1 self)
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Abstract. Fáry’s theorem states that every plane graph can be drawn as a straightline drawing. A plane graph is a graph embedded in a plane without edge crossings. In this paper, we extend Fáry’s theorem to nonplanar graphs. More specifically, we study the problem of drawing 1plane graphs with straightline edges. A 1plane graph is a graph embedded in a plane with at most one crossing per edge. We give a characterisation of those 1plane 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 1plane 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
A tighter insertionbased approximation of the crossing number. Full version. ArXiv
, 2011
"... Abstract. Let G be a planar graph and F a set of additional edges not yet in G. The multiple edge insertion problem (MEI) asks for a drawing of G+F with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. As an exact solution to MEI is NPhard for general F, we pre ..."
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Cited by 1 (1 self)
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Abstract. Let G be a planar graph and F a set of additional edges not yet in G. The multiple edge insertion problem (MEI) asks for a drawing of G+F with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. As an exact solution to MEI is NPhard for general F, we present the first approximation algorithm for MEI which achieves an additive approximation factor (depending only on the size of F and the maximum degree of G) in the case of connected G. Our algorithm seems to be the first directly implementable one in that realm, too, next to the single edge insertion. It is also known that an (even approximate) solution to the MEI problem would approximate the crossing number of the Falmostplanar graph G+F, while computing the crossing number of G+F exactly is NPhard already when F  = 1. Hence our algorithm induces new, improved approximation bounds for the crossing number problem of Falmostplanar graphs, achieving constantfactor approximation for the large class of such graphs of bounded degrees and bounded size of F. 1
Recognizing Outer 1Planar Graphs in Linear Time ⋆,⋆⋆
"... Abstract. A graph is outer 1planar (o1p) if it can be drawn in the plane such that all vertices are on the outer face and each edge is crossed at most once. o1p graphs generalize outerplanar graphs, which can be recognized in linear time and specialize 1planar graphs, whose recognition is N Phard ..."
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Cited by 1 (1 self)
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Abstract. A graph is outer 1planar (o1p) if it can be drawn in the plane such that all vertices are on the outer face and each edge is crossed at most once. o1p graphs generalize outerplanar graphs, which can be recognized in linear time and specialize 1planar graphs, whose recognition is N Phard. Our main result is a lineartime algorithm that first tests whether a graph G is o1p, and then computes an embedding. Moreover, the algorithm can augment G to a maximal o1p graph. If G is not o1p, then it includes one of six minors (see Fig. 3), which are also detected by the recognition algorithm. Hence, the algorithm returns a positive or negative witness for o1p. 1