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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 28 (11 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.
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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Cited by 7 (2 self)
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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.
An SPQRTree Approach to Decide Special Cases of Simultaneous Embedding with Fixed Edges
, 2008
"... We present a lineartime algorithm for solving the simultaneous embedding problem with fixed edges (SEFE) for a planar graph and a pseudoforest (a graph with at most one cycle) by reducing it to the following embedding problem: Given a planar graph G, a cycle C of G, and a partitioning of the remai ..."
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Cited by 5 (0 self)
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We present a lineartime algorithm for solving the simultaneous embedding problem with fixed edges (SEFE) for a planar graph and a pseudoforest (a graph with at most one cycle) by reducing it to the following embedding problem: Given a planar graph G, a cycle C of G, and a partitioning of the remaining vertices of G, does there exist a planar embedding in which the induced subgraph on each vertex partite of G \ C is contained entirely inside or outside C? For the latter problem, we present an algorithm that is based on SPQRtrees and has linear running time. We also show how we can employ SPQRtrees to decide SEFE for two planar graphs where one graph has at most two cycles and the intersection is a pseudoforest in linear time. These results give rise to our hope that our SPQRtree approach might eventually lead to a polynomialtime algorithm for deciding the general SEFE problem for two planar graphs.
Consistent Weighted Graph Layouts
"... Abstract. A graph layout is a geometrical representation of a graph such that the vertexes are assigned points and the edges become line segments. In this paper we present two probabilistic algorithms that build layouts for weighted graphs such that the geometrical distances between the vertexes are ..."
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Abstract. A graph layout is a geometrical representation of a graph such that the vertexes are assigned points and the edges become line segments. In this paper we present two probabilistic algorithms that build layouts for weighted graphs such that the geometrical distances between the vertexes are consistent with the weights of the edges. Both methods start with a random layout and improve it in a number of iterations to decrease the error between the weight of the edges and the length of the corresponding line segments. Both methods have been successful in building consistent layouts with high precision. Mathematics Subject Classification (2000). graph drawing.
Consistent Graph Layout for Weighted Graphs by a ForceBased Probabilistic Algorithm
, 2004
"... In this paper we present three algorithms that build graph layouts for undirected, weighted graphs. Our goal is to generate layouts that are consistent with the weights in the graph. All of the algorithms are forceoriented and have been successful in solving the problem up to a certain precision. T ..."
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In this paper we present three algorithms that build graph layouts for undirected, weighted graphs. Our goal is to generate layouts that are consistent with the weights in the graph. All of the algorithms are forceoriented and have been successful in solving the problem up to a certain precision. They all start with a random layout and improve it by iteratively repositioning the vertices to reduce the current error. The first two methods move the vertices along one edge at a time, either by selecting it randomly, or by following a breadthfirst strategy. The third method computes the result of all of the tension forces occurring in each vertex and moves all of them in each step along the resulting vectors. We also show that if we start building the layout with a robust method and then refine the configuration with a more precise one, we can improve the quality of the solution.