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### Point-Set Embedding of Trees with Edge Constraints [chapter]

Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, Henk Meijer, Stephen Wismath
2008 Lecture Notes in Computer Science
A geometric point-set embedding is a point-set embedding with no edge bends.  ...  Given a graph G with n vertices and a set S of n points in the plane, a point-set embedding of G on S is a planar drawing such that each vertex of G is mapped to a distinct point of S.  ...  Conclusions and Open Problems This paper introduced the problem of computing a point-set embedding of a graph G on a set S of points, with the constraint that a partial straight-line planar drawing of  ...

### Treedecomposition of geometric constraint graphs based on computing graph circuits

Robert Joan-Arinyo, Marta Tarrés-Puertas, Sebastià Vila-Marta
2009 2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling on - SPM '09
First the geometric problem is translated into a graph whose vertices represent the set of geometric elements and whose edges are the constraints.  ...  The algorithm recursively first computes a set of fundamental circuits in the graph then splits the graph into a set of subgraphs each sharing exactly three vertices with the fundamental circuit.  ...  We consider 2D geometric constraint problems defined by a set of geometric elements like points, lines, line segments, circles and circular arcs with fixed radius, along with a set of constraints like  ...

### Page 5896 of Mathematical Reviews Vol. , Issue 98I [page]

1998 Mathematical Reviews
In the rooted-tree embedding problem we are given a rooted tree T with n nodes and a set of n points P with one designated point p and are asked to find a straight-line embedding of T into P with the root  ...  In the degree- constrained embedding problem we are given a set of n points P where each point is assigned a positive degree and the degrees sum to 2m —2 and are asked to embed a tree in P using straight  ...

### Optimizing over All Combinatorial Embeddings of a Planar Graph (Extended Abstract) [chapter]

Petra Mutzel, René Weiskircher
1999 Lecture Notes in Computer Science
Our computational results on two benchmark sets of graphs are surprising: The number of variables and constraints seems to grow only linearly with the size of the graphs although the number of embeddings  ...  We characterize the set of all possible embeddings of a given biconnected planar graph G by means of a system of linear inequalities with f0; 1gvariables corresponding to the set of those cycles in G which  ...  Di Battista in Rome for giving us the opportunity to use their implementation of SPQR-trees in GDToolkit, a software library that is part of the ESPRIT ALCOM-IT project (work package 1.2), and to use their  ...

### A New Perspective on Clustered Planarity as a Combinatorial Embedding Problem [article]

Thomas Bläsius, Ignaz Rutter
2015 arXiv   pre-print
Moreover, the cd-tree reveals interesting connections between c-planarity and planarity with constraints on the order of edges around vertices.  ...  (i) Every cluster and every co-cluster (complement of a cluster) has at most two connected components. (ii) Every cluster has at most five outgoing edges.  ...  We believe that the constrained embedding problems we defined provide a promising starting point for further research, e.g., by studying restricted variants to further deepen the understanding of the c-planarity  ...

### A New Perspective on Clustered Planarity as a Combinatorial Embedding Problem [chapter]

Thomas Bläsius, Ignaz Rutter
2014 Lecture Notes in Computer Science
(ii) Every cluster has at most five outgoing edges. Moreover, the cd-tree reveals interesting connections between c-planarity and planarity with constraints on the order of edges around vertices.  ...  We first present the cd-tree data structure (Sec. 3) and use it to characterize c-planarity in terms of combinatorial embeddings of planar graphs.  ...  A family of order-constraints for the set S is a set of different order constraints, i.e., a subset of the power set of O S .  ...

### Bend Minimization in Planar Orthogonal Drawings Using Integer Programming

Petra Mutzel, René Weiskircher
2006 SIAM Journal on Optimization
Our approach for biconnected graphs combines an integer linear programming (ILP) formulation for the set of all embeddings of a planar graph with the network flow formulation for fixed embeddings.  ...  While the problem can be solved via network flow for a given planar embedding of a graph G, it is NP-hard if we consider the set of all planar embeddings of G.  ...  Acknowledgment We thank Walter Didimo for providing the code of the branch & bound algorithm and the real world benchmark set.  ...

### Crossing Minimization and Layouts of Directed Hypergraphs with Port Constraints [chapter]

Markus Chimani, Carsten Gutwenger, Petra Mutzel, Miro Spönemann, Hoi-Ming Wong
2011 Lecture Notes in Computer Science
Most existing algorithms for drawing hypergraphs with port constraints are adaptions of the framework originally proposed by Sugiyama et al. in 1981 for simple directed graphs.  ...  Many practical applications for drawing graphs are modeled by directed graphs with domain specific constraints.  ...  A generic approach to model this set of orders for each node is by using embedding constraints, which define a tree structure of constraint nodes [12] .  ...

### Simultaneous Embedding of Planar Graphs with Few Bends [chapter]

Cesim Erten, Stephen G. Kobourov
2005 Lecture Notes in Computer Science
We also describe an O(n) time algorithm for simultaneous embedding with fixed-edges for tree-path pairs on the O(n) × O(n 2 ) grid with at most one bend per tree-edge and no bends along path edges.  ...  We present an O(n) time algorithm for simultaneous embedding of pairs of planar graphs on the O(n 2 )×O(n 2 ) grid, with at most three bends per edge, where n is the number of vertices.  ...  Acknowledgments We would like to thank Michael Kaufmann and David Eppstein for the interesting discussions about the problems discussed in the paper, and Petr Moravsky for helping with the implementation  ...

### Directional Routing via Generalized st-Numberings

Fred S. Annexstein, Kenneth A. Berman
2000 SIAM Journal on Discrete Mathematics
We also show that a directed acyclic graph (dag) that is k-connected to a set of sinks has a k-directional embedding in (k − 1)-space with the sink set as the extreme nodes.  ...  We prove that all 3-connected graphs have 3-directional embeddings in the plane so that each node outside a set of extreme nodes has a neighbor in each of the three directional regions defined in the plane  ...  We now show that it is still possible to "split" a set of three points in the plane into three directions even with the addition of many new directional constraints of a certain type. Proof.  ...

### Testing Simultaneous Planarity when the Common Graph is 2-Connected [article]

Bernhard Haeupler and Krishnam Raju Jampani and Anna Lubiw
2011 arXiv   pre-print
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.  ...  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  ...  At any point we will have two sets of PQ-trees representing the partial planar embeddings of the subgraphs of G 1 and G 2 induced by the vertices added so far.  ...

### Algorithms for Metric Learning via Contrastive Embeddings

Diego Ihara, Neshat Mohammadi, Anastasios Sidiropoulos, Michael Wagner
2019 International Symposium on Computational Geometry
Given a universe X and sets S, D ⊂ X 2 of similar and dissimilar pairs, we seek to find a mapping f : X → Y , into some target metric space M = (Y, ρ), such that similar objects are mapped to points at  ...  We study the problem of supervised learning a metric space under discriminative constraints.  ...  Let G S be the graph with vertex set V (G S ) = X, and edge set E(G S ) = S. We use ρ S to denote the shortest-path distance of G S , where the length of each edge is set to u. Lemma 10.  ...

### Testing Simultaneous Planarity When the Common Graph Is 2-Connected [chapter]

Bernhard Haeupler, Krishnam Raju Jampani, Anna Lubiw
2010 Lecture Notes in Computer Science
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  ...  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 2-connected.  ...  In particular we want to thank one referee for help with the drawing method in Section 4.3.  ...

### Graph Embedding with Topological Cycle-Constraints [chapter]

Christoph Dornheim
1999 Lecture Notes in Computer Science
We address the problem of nding a planar embedding of a graph satisfying a set of constraints between its vertices and cycles that require embedding a given vertex inside its corresponding cycle.  ...  This paper concerns graph embedding under topological constraints.  ...  Thus, we are concerned with the problem of nding a planar embedding of a graph satisfying a given set of cycle-constraints.  ...

### Simultaneous Embedding of Planar Graphs with Few Bends

Cesim Erten, Stephen G. Kobourov
2005 Journal of Graph Algorithms and Applications
We also describe an O(n) time algorithm for simultaneous embedding with fixed-edges for tree-path pairs on the O(n) × O(n 2 ) grid with at most one bend per tree-edge and no bends along path edges.  ...  We present an O(n) time algorithm for simultaneous embedding of pairs of planar graphs on the O(n 2 )×O(n 2 ) grid, with at most three bends per edge, where n is the number of vertices.  ...  Acknowledgments We would like to thank Michael Kaufmann and David Eppstein for the interesting discussions about the problems discussed in the paper, and Petr Moravsky for helping with the implementation  ...
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