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60
Planarity testing and optimal edge insertion with embedding constraints
, 2008
"... The planarization method has proven to be successful in graph drawing. The output, a combinatorial planar embedding of the socalled planarized graph, can be combined with stateoftheart planar drawing algorithms. However, many practical applications have additional constraints on the drawings tha ..."
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The planarization method has proven to be successful in graph drawing. The output, a combinatorial planar embedding of the socalled planarized graph, can be combined with stateoftheart planar drawing algorithms. However, many practical applications have additional constraints on the drawings that result in restrictions on the set of admissible planar embeddings. In this paper, we consider embedding constraints that restrict the admissible order of incident edges around a vertex. Such constraints occur in applications, e.g., from side or port constraints. We introduce a set of hierarchical embedding constraints that include grouping, oriented, and mirror constraints, and show how these constraints can be integrated into the planarization method. For this, we first present a linear time algorithm for testing if a given graph G is ecplanar, i.e., admits a planar embedding satisfying the given embedding constraints. In the case that G is ecplanar, we provide a linear time algorithm for computing the corresponding ecembedding. Otherwise, an ecplanar subgraph is computed. The critical part is to reinsert the deleted edges subject to the embedding constraints so that the number of crossings is kept small. For this, we present a linear time algorithm which is able to insert an edge into an ecplanar graph H so that the insertion is crossing minimal among all ecplanar embeddings of H. As a side result, we characterize the set of all possible ecplanar embeddings using BC and SPQRtrees.
Unveiling Hidden Unstructured Regions in Process Models
 In OTM, volume 5870 of LNCS
, 2009
"... Abstract. Process models define allowed process execution scenarios. The models are usually depicted as directed graphs, with gateway nodes regulating the control flow routing logic and with edges specifying the execution order constraints between tasks. While arbitrarily structured control flow pat ..."
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Abstract. Process models define allowed process execution scenarios. The models are usually depicted as directed graphs, with gateway nodes regulating the control flow routing logic and with edges specifying the execution order constraints between tasks. While arbitrarily structured control flow patterns in process models complicate model analysis, they also permit creativity and full expressiveness when capturing nontrivial process scenarios. This paper gives a classification of arbitrarily structured process models based on the hierarchical process model decomposition technique. We identify a structural class of models consisting of block structured patterns which, when combined, define complex execution scenarios spanning across the individual patterns. We show that complex behavior can be localized by examining structural relations of loops in hidden unstructured regions of control flow. The correctness of the behavior of process models within these regions can be validated in linear time. These observations allow us to suggest techniques for transforming hidden unstructured regions into blockstructured ones.
Recognizing Outer 1Planar Graphs in Linear Time
"... 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 mai ..."
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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.
CPlanarity of cconnected clustered graphs
, 2008
"... We present the first characterization of cplanarity for cconnected clustered graphs. The characterization is based on the interplay between the hierarchy of the clusters and the hierarchies of the triconnected and biconnected components of the underlying graph. Based on such a characterization, we ..."
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Cited by 6 (4 self)
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We present the first characterization of cplanarity for cconnected clustered graphs. The characterization is based on the interplay between the hierarchy of the clusters and the hierarchies of the triconnected and biconnected components of the underlying graph. Based on such a characterization, we provide a lineartime cplanarity testing and embedding algorithm for cconnected clustered graphs. The algorithm is reasonably easy to implement, since it exploits as building blocks simple algorithmic tools like the computation of lowest common ancestors, minimum and maximum spanning trees, and counting sorts. It also makes use of wellknown data structures as SPQRtrees and BCtrees. If the test fails, the algorithm identifies a structural element responsible for the noncplanarity of the input clustered graph.
A Simple Algorithm for Triconnectivity of a Multigraph
, 2009
"... Vertexconnectivity and edgeconnectivity represent the extent to which a graph is connected. Study of these key properties of graphs plays an important role in varieties of computer science applications. Recent years have witnessed a number of linear time 3edgeconnectivity algorithms with increa ..."
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Cited by 5 (3 self)
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Vertexconnectivity and edgeconnectivity represent the extent to which a graph is connected. Study of these key properties of graphs plays an important role in varieties of computer science applications. Recent years have witnessed a number of linear time 3edgeconnectivity algorithms with increasing simplicity. In contrast, the stateoftheart algorithm for 3vertexconnectivity due to Hopcroft and Tarjan lacks the simplicity in the sense of ease of implementation as well as the number of passes over the graph although its time and space complexity is theoretically linear. In this paper, we propose a linear time reduction from 3vertexconnectivity to 3edgeconnectivity of a multigraph. This reduction was previously unknown, while the reduction in the opposite direction already exists. Applying an existing linear time 3edgeconnectivity algorithm on the reduced graph will solve the 3vertexconnectivity problem of the original graph. Hence, for a graph with V vertices and E  edges, the proposed reduction turns into an O(V + E) time and space algorithm for 3vertexconnectivity while enjoying the simplicity of the 3edgeconnectivity algorithms.
Monotone Drawings of Graphs
, 2012
"... We study a new standard for visualizing graphs: A monotone drawing is a straightline drawing such that, for every pair of vertices, there exists a path that monotonically increases with respect to some direction. We show algorithms for constructing monotone planar drawings of trees and biconnected ..."
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We study a new standard for visualizing graphs: A monotone drawing is a straightline drawing such that, for every pair of vertices, there exists a path that monotonically increases with respect to some direction. We show algorithms for constructing monotone planar drawings of trees and biconnected planar graphs, we study the interplay between monotonicity, planarity, and convexity, and we outline a number of open problems and future research directions.
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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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
I.: Disconnectivity and relative positions in simultaneous embeddings
, 2013
"... The problem Simultaneous Embedding with Fixed Edges (SEFE) asks for two planar ..."
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The problem Simultaneous Embedding with Fixed Edges (SEFE) asks for two planar
Progress on Certifying Algorithms
"... A certifying algorithm is an algorithm that produces with each output, a certificate or witness (easytoverify proof) that the particular output has not been compromised by a bug. A user of a certifying program P ( = the implementation of a certifying algorithm) inputs x, receives an output y and a ..."
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A certifying algorithm is an algorithm that produces with each output, a certificate or witness (easytoverify proof) that the particular output has not been compromised by a bug. A user of a certifying program P ( = the implementation of a certifying algorithm) inputs x, receives an output y and a certificate w, and then checks, either manually or by use of a checking program, that w proves that y is a correct output for input x. In this way, he/she can be sure of the correctness of the output without having to trust P. We refer the reader to the recent survey paper [9] for a detailed discussion of certifying algorithms. 1 An Example We illustrate the concept by an example. A matching in a graph G is a subset M of the edges of G such that no two share an endpoint. A matching has maximum cardinality if its cardinality is at least as large as that of any other matching. Figure 1 shows a graph and a maximum cardinality matching. Observe that the matching leaves two nodes unmatched, which gives rise to the question whether there exists a matching of larger cardinality. What is a witness for a matching being of maximum cardinality? Edmonds in his seminal papers [1,2] on how to compute maximum matchings in polynomial time introduced the following certificate: An oddset cover OSC of G is
NonPlanar Core Reduction of Graphs
"... We present a reduction method that reduces a graph to a smaller core graph which behaves invariant with respect to planarity measures like crossing number, skewness, and thickness. The core reduction is based on the decomposition of a graph into its triconnected components and can be computed in l ..."
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We present a reduction method that reduces a graph to a smaller core graph which behaves invariant with respect to planarity measures like crossing number, skewness, and thickness. The core reduction is based on the decomposition of a graph into its triconnected components and can be computed in linear time. It has applications in heuristic and exact optimization algorithms for the planarity measures mentioned above. Experimental results show that this strategy yields a reduction to 2/3 in average for a widely used benchmark set of graphs.