
Kovacs, Robert; Seufert, Anna; Wall, Ludwig; Chen, HsiangTing; Meinel, Florian; Müller, Willi; You, Sijing; Brehm, Maximilian; Striebel, Jonathan; Kommana, Yannis; Popiak, Alexander; Bläsius, Thomas; Baudisch, Patrick TrussFab: Fabricating Sturdy LargeScale Structures on Desktop 3D Printers. Human Factors in Computing Systems (CHI) 2017: 26062616
We present TrussFab, an integrated endtoend system that allows users to fabricate large scale structures that are sturdy enough to carry human weight. TrussFab achieves the large scale by complementing 3D print with plastic bottles. It does not use these bottles as "bricks" though, but as beams that form structurally sound nodelink structures, also known as trusses, allowing it to handle the forces resulting from scale and load. TrussFab embodies the required engineering knowledge, allowing nonengineers to design such structures and to validate their design using integrated structural analysis. We have used TrussFab to design and fabricate tables and chairs, a 2.5 m long bridge strong enough to carry a human, a functional boat that seats two, and a 5 m diameter dome.

Bläsius, Thomas; Radermacher, Marcel; Rutter, Ignaz How to Draw a Planarization. Software Seminar (SOFSEM) 2017: 295308
We study the problem of computing straightline drawings of nonplanar graphs with few crossings. We assume that a crossingminimization algorithm is applied first, yielding a planarization, i.e., a planar graph with a dummy vertex for each crossing, that fixes the topology of the resulting drawing. We present and evaluate two different approaches for drawing a planarization in such a way that the edges of the input graph are as straight as possible. The first approach is based on the planaritypreserving forcedirected algorithm ImPrEd, the second approach, which we call Geometric Planarization Drawing, iteratively moves vertices to their locally optimal positions in the given initial drawing.

Bläsius, Thomas; Friedrich, Tobias; Krohmer, Anton Cliques in Hyperbolic Random Graphs. Algorithmica 2017
Most complex real world networks display scalefree features. This characteristic motivated the study of numerous random graph models with a powerlaw degree distribution. There is, however, no established and simple model which also has a high clustering of vertices as typically observed in real data. Hyperbolic random graphs bridge this gap. This natural model has recently been introduced by hyprg and has shown theoretically and empirically to fulfill all typical properties of real world networks, including powerlaw degree distribution and high clustering. We study cliques in hyperbolic random graphs \(G\) and present new results on the expected number of \(k\)cliques \(E(K_k)\) and the size of the largest clique \(\omega(G)\). We observe that there is a phase transition at powerlaw exponent \(beta = 3\). More precisely, for \(beta in (2,3)\) we prove \(E(K_k)=n^{k (3\beta)/2} \Theta(k)^{k}\) and \(\omega(G)=\Theta(n^{(3\beta)/2})\), while for \(\beta\geq3\) we prove \(E(K_k)=n \Theta(k)^{k}\) and \(\omega(G)=\Theta(\log(n)/ log log n)\). Furthermore, we show that for \(beta geq 3\), cliques in hyperbolic random graphs can be computed in time \(O(n)\). If the underlying geometry is known, cliques can be found with worstcase runtime \(O(m cdot n^{2.5})\) for all values of \(\beta\).

Bläsius, Thomas; Friedrich, Tobias; Krohmer, Anton Hyperbolic Random Graphs: Separators and Treewidth. European Symposium on Algorithms (ESA) 2016: 15:115:16
When designing and analyzing algorithms, one can obtain better and more realistic results for practical instances by assuming a certain probability distribution on the input. The worstcase runtime is then replaced by the expected runtime or by bounds that hold with high probability (whp), i.e., with probability \(1 neq O(1/n)\), on the random input. Hyperbolic random graphs can be used to model complex realworld networks as they share many important properties such as a small diameter, a large clustering coefficient, and a powerlaw degreedistribution. Divide and conquer is an important algorithmic design principle that works particularly well if the instance admits small separators. We show that hyperbolic random graphs in fact have comparatively small separators. More precisely, we show that a hyperbolic random graph can be expected to have a balanced separator hierarchy with separators of size \(O(\sqrt(n^{(3\beta)}))\), \(O(log n)\), and \(O(1)\) if \(2 < beta < 3\), \(beta = 3\) and \(3 < \beta\), respectively (\(\beta\) is the powerlaw exponent). We infer that these graphs have whp a treewidth of \(O(\sqrt(n^{(3  \beta)}))\), \(O(\log^{2}n)\), and \(O(log n)\), respectively. For \(2 < beta < 3\), this matches a known lower bound. For the more realistic (but harder to analyze) binomial model, we still prove a sublinear bound on the treewidth. To demonstrate the usefulness of our results, we apply them to obtain fast matching algorithms and an approximation scheme for Independent Set.

Bläsius, Thomas; Rutter, Ignaz A new perspective on clustered planarity as a combinatorial embedding problem. Theoretical Computer Science 2016: 306315
The clustered planarity problem (cplanarity) asks whether a hierarchically clustered graph admits a planar drawing such that the clusters can be nicely represented by regions. We introduce the cdtree data structure and give a new characterization of cplanarity. It leads to efficient algorithms for cplanarity testing in the following cases. (i) Every cluster and every cocluster (complement of a cluster) has at most two connected components. (ii) Every cluster has at most five outgoing edges. Moreover, the cdtree reveals interesting connections between cplanarity and planarity with constraints on the order of edges around vertices. On one hand, this gives rise to a bunch of new open problems related to cplanarity, on the other hand it provides a new perspective on previous results.

Bläsius, Thomas; Friedrich, Tobias; Schirneck, Martin The Parameterized Complexity of Dependency Detection in Relational Databases. International Symposium on Parameterized and Exact Computation (IPEC) 2016: 6:16:13
We study the parameterized complexity of classical problems that arise in the profiling of relational data. Namely, we characterize the complexity of detecting unique column combinations (candidate keys), functional dependencies, and inclusion dependencies with the solution size as parameter. While the discovery of uniques and functional dependencies, respectively, turns out to be W[2]complete, the detection of inclusion dependencies is one of the first natural problems proven to be complete for the class W[3]. As a side effect, our reductions give insights into the complexity of enumerating all minimal unique column combinations or functional dependencies.

Bläsius, Thomas; Friedrich, Tobias; Krohmer, Anton; Laue, Sören Efficient Embedding of ScaleFree Graphs in the Hyperbolic Plane. European Symposium on Algorithms (ESA) 2016: 16:116:18
EATCS Best Paper Award
Hyperbolic geometry appears to be intrinsic in many large real networks. We construct and implement a new maximum likelihood estimation algorithm that embeds scalefree graphs in the hyperbolic space. All previous approaches of similar embedding algorithms require a runtime of \(\Omega(n^{2})\). Our algorithm achieves quasilinear runtime, which makes it the first algorithm that can embed networks with hundreds of thousands of nodes in less than one hour. We demonstrate the performance of our algorithm on artificial and real networks. In all typical metrics like Loglikelihood and greedy routing our algorithm discovers embeddings that are very close to the ground truth.

Bläsius, Thomas; Rutter, Ignaz; Wagner, Dorothea Optimal Orthogonal Graph Drawing with Convex Bend Costs. Transactions on Algorithms 2016: 33
Traditionally, the quality of orthogonal planar drawings is quantified by the total number off bends or the maximum number of bends per edge. However, this neglects that, in typical applications, edges have varying importance. We consider the problem OptimalFlexDraw that is defined as follows. Given a planar graph \(G\) on \(n\) vertices with maximum degree 4 (4planar graph) and for each edge \(e\) a cost function \(cost_e colon N_0 rightarrow R\) defining costs depending on the number of bends \(e\) has, compute a planar orthogonal drawing of \(G\) of minimum cost. In this generality OptimalFlexDraw is NPhard. We show that it can be solved efficiently if (1) the cost function of each edge is convex and (2) the first bend on each edge does not cause any cost. Our algorithm takes time \(O(n, cdot, T_{flow}(n)\) and \(O(n^2, cdot, T_{flow}(n))\) for biconnected and connected graphs, respectively, where \(T_{flow}(n)\) denotes the time to compute a minimumcost flow in a planar network with multiple sources and sinks. Our result is the first polynomialtime bendoptimization algorithm for general 4planar graphs optimizing over all embeddings. Previous work considers restricted graph classes and unit costs.

Bläsius, Thomas; Lehmann, Sebastian; Rutter, Ignaz Orthogonal Graph Drawing with Inflexible Edges. Computational Geometry 2016: 2640
We consider the problem of creating plane orthogonal drawings of 4planar graphs (planar graphs with maximum degree 4) with constraints on the number of bends per edge. More precisely, we have a flexibility function assigning to each edge \(e\) a natural number \(flex(e)\), its flexibility. The problem FlexDraw asks whether there exists an orthogonal drawing such that each edge \(e\) has at most \(flex(e)\) bends. It is known that FlexDraw is NPhard if \(flex(e)=0\) for every edge \(e\) 7. On the other hand, FlexDraw can be solved efficiently if \(flex(e) ge 1\) 2 and is trivial if \(flex(e) ge 2\) 1 for every edge \(e\). To close the gap between the NPhardness for \(flex(e)=0\) and the efficient algorithm for \(flex(e) ge 1\), we investigate the computational complexity of FlexDraw in case only few edges are inflexible (i.e., have flexibility 0). We show that for any \(epsilon > 0\) FlexDraw is NPcomplete for instances with \(O(n^\epsilon)\) inflexible edges with pairwise distance \(\Omega(n^{1\epsilon})\) (including the case where they induce a matching). On the other hand, we give an FPTalgorithm with running time \(O(2^k cdot n cdot T_{flow}(n))\), where \(T_{flow}(n)\) is the time necessary to compute a maximum flow in a planar flow network with multiple sources and sinks, and \(k\) is the number of inflexible edges having at least one endpoint of degree 4.

Bläsius, Thomas; Rutter, Ignaz Simultaneous PQOrdering with Applications to Constrained Embedding Problems. Transactions on Algorithms 2016: 16
In this article, we define and study the new problem of SIMULTANEOUS PQORDERING. Its input consists of a set of PQtrees, which represent sets of circular orders of their leaves, together with a set of childparent relations between these PQtrees, such that the leaves of the child form a subset of the leaves of the parent. SIMULTANEOUS PQORDERING asks whether orders of the leaves of each of the trees can be chosen simultaneously; that is, for every childparent relation, the order chosen for the parent is an extension of the order chosen for the child. We show that SIMULTANEOUS PQORDERING is NPcomplete in general, and we identify a family of instances that can be solved efficiently, the 2fixed instances. We show that this result serves as a framework for several other problems that can be formulated as instances of SIMULTANEOUS PQORDERING. In particular, we give lineartime algorithms for recognizing simultaneous interval graphs and extending partial interval representations. Moreover, we obtain a lineartime algorithm for PARTIALLY PQCONSTRAINED PLANARITY for biconnected graphs, which asks for a planar embedding in the presence of 16 PQtrees that restrict the possible orderings of edges around vertices, and a quadratictime algorithm for SIMULTANEOUS EMBEDDING WITH FIXED EDGES for biconnected graphs with a connected intersection. Both results can be extended to the case where the input graphs are not necessarily biconnected but have the property that each cutvertex is contained in at most two nontrivial blocks. This includes, for example, the case where both graphs have a maximum degree of 5.

Baum, Moritz; Bläsius, Thomas; Gemsa, Andreas; Rutter, Ignaz; Wegner, Franziska Scalable Exact Visualization of Isocontours in Road Networks via MinimumLink Paths. European Symposium on Algorithms (ESA) 2016: 7:17:18
Isocontours in road networks represent the area that is reachable from a source within a given resource limit. We study the problem of computing accurate isocontours in realistic, largescale networks. We propose isocontours represented by polygons with minimum number of segments that separate reachable and unreachable components of the network. Since the resulting problem is not known to be solvable in polynomial time, we introduce several heuristics that run in (almost) linear time and are simple enough to be implemented in practice. A key ingredient is a new practical lineartime algorithm for minimumlink paths in simple polygons. Experiments in a challenging realistic setting show excellent performance of our algorithms in practice, computing nearoptimal solutions in a few milliseconds on average, even for long ranges.

Bläsius, Thomas New Approaches to Classic GraphEmbedding Problems  Orthogonal Drawings & Constrained Planarity. 2015
Drawings of graphs are often used to represent a given data set in a humanreadable way. In this thesis, we consider different classic algorithmic problems that arise when automatically generating graph drawings. More specifically, we solve some open problems in the context of orthogonal drawings and advance the current state of research on the problems clustered planarity and simultaneous planarity.

Alam, Md. Jawaherul; Bläsius, Thomas; Rutter, Ignaz; Ueckerdt, Torsten; Wolff, Alexander Pixel and Voxel Representations of Graphs. Graph Drawing (GD) 2015: 472486
We study contact representations for graphs, which we call pixel representations in 2D and voxel representations in 3D. Our representations are based on the unit square grid whose cells we call pixels in 2D and voxels in 3D. Two pixels are adjacent if they share an edge, two voxels if they share a face. We call a connected set of pixels or voxels a blob. Given a graph, we represent its vertices by disjoint blobs such that two blobs contain adjacent pixels or voxels if and only if the corresponding vertices are adjacent. We are interested in the size of a representation, which is the number of pixels or voxels it consists of. We first show that finding minimumsize representations is NPcomplete. Then, we bound representation sizes needed for certain graph classes. In 2D, we show that, for \(k\)outerplanar graphs with \(n\) vertices, \(\Theta(kn)\) pixels are always sufficient and sometimes necessary. In particular, outerplanar graphs can be represented with a linear number of pixels, whereas general planar graphs sometimes need a quadratic number. In 3D, \(\Theta(n^2)\) voxels are always sufficient and sometimes necessary for any \(n\)vertex graph. We improve this bound to \(\Theta(n cdot \tau)\) for graphs of treewidth \(\tau\) and to \(O((g+1)^2 n log^2 n)\) for graphs of genus \(g\). In particular, planar graphs admit representations with \(O(nlog^2 n)\) voxels.

Bläsius, Thomas; Rutter, Ignaz Disconnectivity and relative positions in simultaneous embeddings. Computational Geometry 2015: 459478
For two planar graphs \(G^1 = ( V^1, E^1)\) and \(G^2 = ( V^2, E^2)\) sharing a common subgraph \(G = G^1 cap G^2\) the problem Simultaneous Embedding with Fixed Edges (SEFE) asks whether they admit planar drawings such that the common graph is drawn the same. Previous algorithms only work for cases where \(G\) is connected, and hence do not need to handle relative positions of connected components. We consider the problem where \(G\), \(G^1\) and \(G^2\) are not necessarily connected.First, we show that a general instance of SEFE can be reduced in linear time to an equivalent instance where \(V^1 = V^2\) and \(G^1\) and \(G^2\) are connected. Second, for the case where \(G\) consists of disjoint cycles, we introduce the CCtree which represents all embeddings of \(G\) that extend to planar embeddings of \(G^1\). We show that CCtrees can be computed in linear time, and that their intersection is again a CCtree. This yields a lineartime algorithm for SEFE if all \(k\) input graphs (possibly \(k > 2\)) pairwise share the same set of disjoint cycles. These results, including the CCtree, extend to the case where \(G\) consists of arbitrary connected components, each with a fixed planar embedding on the sphere. Then the running time is \(O(n^2)\).

Bläsius, Thomas; Lehmann, Sebastian; Rutter, Ignaz Orthogonal Graph Drawing with Inflexible Edges. Conference on Algorithms and Complexity (CIAC) 2015: 6173
We consider the problem of creating plane orthogonal drawings of 4planar graphs (planar graphs with maximum degree 4) with constraints on the number of bends per edge. More precisely, we have a flexibility function assigning to each edge \(e\) a natural number \(flex(e)\), its flexibility. The problem FlexDraw asks whether there exists an orthogonal drawing such that each edge \(e\) has at most \(flex(e)\) bends. It is known that FlexDraw is NPhard if \(flex(e)=0\) for every edge \(e\) [7]. On the other hand, FlexDraw can be solved efficiently if \(flex(e) \ge1\) [2] and is trivial if \(flex(e) ge 2\) [1] for every edge \(e\). To close the gap between the NPhardness for \(flex(e)=0\) and the efficient algorithm for \(flex(e) ge 1\), we investigate the computational complexity of FlexDraw in case only few edges are inflexible (i.e., have flexibility 0). We show that for any \(epsilon > 0\) FlexDraw is NPcomplete for instances with \(O(n^\epsilon)\) inflexible edges with pairwise distance \(\Omega(n^{1\epsilon})\) (including the case where they induce a matching). On the other hand, we give an FPTalgorithm with running time \(O(2^k cdot n cdot T_{flow}(n))\), where \(T_{flow}(n)\) is the time necessary to compute a maximum flow in a planar flow network with multiple sources and sinks, and \(k\) is the number of inflexible edges having at least one endpoint of degree 4.

Bläsius, Thomas; Rutter, Ignaz A New Perspective on Clustered Planarity as a Combinatorial Embedding Problem. Graph Drawing (GD) 2014: 440451
The clustered planarity problem (\(c\)planarity) asks whether a hierarchically clustered graph admits a planar drawing such that the clusters can be nicely represented by regions. We introduce the \(cd\)tree data structure and give a new characterization of \(c\)planarity. It leads to efficient algorithms for \(c\)planarity testing in the following cases. (i) Every cluster and every cocluster has at most two connected components. (ii) Every cluster has at most five outgoing edges. Moreover, the cdtree reveals interesting connections between \(c\)planarity and planarity with constraints on the order of edges around vertices. On one hand, this gives rise to a bunch of new open problems related to \(c\)planarity, on the other hand it provides a new perspective on previous results.

Angelini, Patrizio; Bläsius, Thomas; Rutter, Ignaz Testing Mutual duality of Planar graphs. Computational Geometry and Applications 2014: 325346
We introduce and study the problem Mutual Planar Duality, which asks for planar graphs \(G_1\) and \(G_2\) whether \(G_1\) can be embedded such that its dual is isomorphic to \(G_2\). We show NPcompleteness for general graphs and give a lineartime algorithm for biconnected graphs. We consider the common dual relation ~, where \(G_1\) ~ \(G_2\) if and only they admit embeddings that result in the same dual graph. We show that ~ is an equivalence relation on the set of biconnected graphs and devise a succinct, SPQRtreelike representation of its equivalence classes. To solve Mutual Planar Duality for biconnected graphs, we show how to do isomorphism testing for two such representations in linear time. A special case of Mutual Planar Duality is testing whether a graph is selfdual. Our algorithm can handle the case of biconnected graphs in linear time and our NPhardness proof extends to selfduality and also to map selfduality testing (which additionally requires to preserve the embedding).

Bläsius, Thomas; Krug, Marcus; Rutter, Ignaz; Wagner, Dorothea Orthogonal Graph Drawing with Flexibility Constraints. Algorithmica 2014: 859885
Traditionally, the quality of orthogonal planar drawings is quantified by either the total number of bends, or the maximum number of bends per edge. However, this neglects that in typical applications, edges have varying importance. In this work, we investigate an approach that allows to specify the maximum number of bends for each edge individually, depending on its importance. We consider a new problem called FlexDraw that is defined as follows. Given a planar graph \(G=(V,E)\) on \(n\) vertices with maximum degree 4 and a function \(flex colon E rightarrow N_0\) that assigns a flexibility to each edge, does \(G\) admit a planar embedding on the grid such that each edge \(e\) has at most \(flex(e)\) bends? Note that in our setting the combinatorial embedding of \(G\) is not fixed. FlexDraw directly extends the problem \(\beta\)embeddability asking whether \(G\) can be embedded with at most \(\beta\) bends per edge. We give an algorithm with runningtime \(O(n^2)\) solving FlexDraw when the flexibility of each edge is positive. This includes 1embeddability as a special case and thus closes the complexity gap between 0embeddability, which is NPhard to decide, and 2embeddability, which is efficiently solvable since every planar graph with maximum degree 4 admits a 2embedding except for the octahedron. In addition to the polynomialtime algorithm we show that FlexDraw is NPhard even if the edges with flexibility 0 induce a tree or a union of disjoint stars.

Bläsius, Thomas; Brückner, Guido; Rutter, Ignaz Complexity of HigherDegree Orthogonal Graph Embedding in the Kandinsky Model. European Symposium on Algorithms (ESA) 2014: 161172
We show that finding orthogonal gridembeddings of plane graphs (planar with fixed combinatorial embedding) with the minimum number of bends in the socalled Kandinsky model (which allows vertices of degree >4) is NPcomplete, thus solving a longstanding open problem. On the positive side, we give an efficient algorithm for several restricted variants, such as graphs of bounded branch width and a subexponential exact algorithm for general plane graphs.

Biedl, Therese C.; Bläsius, Thomas; Niedermann, Benjamin; Nöllenburg, Martin; Prutkin, Roman; Rutter, Ignaz Using ILP/SAT to Determine Pathwidth, Visibility Representations, and other GridBased Graph Drawings. Graph Drawing (GD) 2013: 460471
We present a simple and versatile formulation of gridbased graph representation problems as an integer linear program (ILP) and a corresponding SAT instance. In a gridbased representation vertices and edges correspond to axisparallel boxes on an underlying integer grid; boxes can be further constrained in their shapes and interactions by additional problemspecific constraints. We describe a general \(d\)dimensional model for grid representation problems. This model can be used to solve a variety of NPhard graph problems, including pathwidth, bandwidth, optimum storientation, areaminimal (bark) visibility representation, boxicityk graphs and others. We implemented SATmodels for all of the above problems and evaluated them on the Rome graphs collection. The experiments show that our model successfully solves NPhard problems within few minutes on small to mediumsize Rome graphs.

Bläsius, Thomas; Rutter, Ignaz Simultaneous PQOrdering with Applications to Constrained Embedding Problems. Symposium on Discrete Algorithms (SODA) 2013: 10301043
In this article, we define and study the new problem of SIMULTANEOUS PQORDERING. Its input consists of a set of PQtrees, which represent sets of circular orders of their leaves, together with a set of childparent relations between these PQtrees, such that the leaves of the child form a subset of the leaves of the parent. SIMULTANEOUS PQORDERING asks whether orders of the leaves of each of the trees can be chosen simultaneously; that is, for every childparent relation, the order chosen for the parent is an extension of the order chosen for the child. We show that SIMULTANEOUS PQORDERING is NPcomplete in general, and we identify a family of instances that can be solved efficiently, the 2fixed instances. We show that this result serves as a framework for several other problems that can be formulated as instances of SIMULTANEOUS PQORDERING. In particular, we give lineartime algorithms for recognizing simultaneous interval graphs and extending partial interval representations. Moreover, we obtain a lineartime algorithm for PARTIALLY PQCONSTRAINED PLANARITY for biconnected graphs, which asks for a planar embedding in the presence of 16 PQtrees that restrict the possible orderings of edges around vertices, and a quadratictime algorithm for SIMULTANEOUS EMBEDDING WITH FIXED EDGES for biconnected graphs with a connected intersection. Both results can be extended to the case where the input graphs are not necessarily biconnected but have the property that each cutvertex is contained in at most two nontrivial blocks. This includes, for example, the case where both graphs have a maximum degree of 5.

Bläsius, Thomas; Karrer, Annette; Rutter, Ignaz Simultaneous Embedding: Edge Orderings, Relative Positions, Cutvertices. Graph Drawing (GD) 2013: 220231
A simultaneous embedding (with fixed edges) of two graphs \(G^1\) and \(G^2\) with common graph \(G = G^1 cap G^2\) is a pair of planar drawings of \(G^1\) and \(G^2\) that coincide on \(G\). It is an open question whether there is a polynomialtime algorithm that decides whether two graphs admit a simultaneous embedding (problem Sefe). In this paper, we present two results. First, a set of three lineartime preprocessing algorithms that remove certain substructures from a given Sefe instance, producing a set of equivalent Sefe instances without such substructures. The structures we can remove are (1) cutvertices of the union graph \(G^{\cup} = G ^1 cup G^2\) , (2) most separating pairs of \(G^{\cup}\), and (3) connected components of \(G\) that are biconnected but not a cycle. Second, we give an \(O(n^3)\)time algorithm solving Sefe for instances with the following restriction. Let \(u\) be a pole of a \(P\)node \(\mu\) in the SPQRtree of a block of \(G^1\) or \(G^2\) . Then at most three virtual edges of \(\mu\) may contain common edges incident to \(u\). All algorithms extend to the sunflower case, i.e., to the case of more than three graphs pairwise intersecting in the same common graph.

Bläsius, Thomas; Rutter, Ignaz; Wagner, Dorothea Optimal Orthogonal Graph Drawing with Convex Bend Costs. International Colloquium on Automata, Languages, and Programming (ICALP) 2013: 184195
Traditionally, the quality of orthogonal planar drawings is quantified by the total number of bends or the maximum number of bends per edge. However, this neglects that, in typical applications, edges have varying importance. We consider the problem OptimalFlexDraw that is defined as follows. Given a planar graph \(G\) on \(n\) vertices with maximum degree 4 (4planar graph) and for each edge \(e\) a cost function \(cost_e colon N_0 rightarrow R\) defining costs depending on the number of bends \(e\) has, compute a planar orthogonal drawing of \(G\) of minimum cost. In this generality OptimalFlexDraw is NPhard. We show that it can be solved efficiently if (1) the cost function of each edge is convex and (2) the first bend on each edge does not cause any cost. Our algorithm takes time \(O(n, cdot, T_{flow}(n)\) and \(O(n^2, cdot, T_{flow}(n))\) for biconnected and connected graphs, respectively, where \(T_{flow}(n)\) denotes the time to compute a minimumcost flow in a planar network with multiple sources and sinks. Our result is the first polynomialtime bendoptimization algorithm for general 4planar graphs optimizing over all embeddings. Previous work considers restricted graph classes and unit costs.

Angelini, Patrizio; Bläsius, Thomas; Rutter, Ignaz Testing Mutual Duality of Planar Graphs. International Symposium on Algorithms and Computation (ISAAC) 2013: 350360
We introduce and study the problem Mutual Planar Duality, which asks for planar graphs \(G_1\) and \(G_2\) whether \(G_1\) can be embedded such that its dual is isomorphic to \(G_2\). We show NPcompleteness for general graphs and give a lineartime algorithm for biconnected graphs. We consider the common dual relation ~, where \(G_1\) ~ \(G_2\) if and only they admit embeddings that result in the same dual graph. We show that ~ is an equivalence relation on the set of biconnected graphs and devise a succinct, SPQRtreelike representation of its equivalence classes. To solve Mutual Planar Duality for biconnected graphs, we show how to do isomorphism testing for two such representations in linear time. A special case of Mutual Planar Duality is testing whether a graph is selfdual. Our algorithm can handle the case of biconnected graphs in linear time and our NPhardness proof extends to selfduality and also to map selfduality testing (which additionally requires to preserve the embedding).

Bläsius, Thomas; Kobourov, Stephen G.; Rutter, Ignaz Simultaneous Embedding of Planar Graphs. Handbook of Graph Drawing and Visualization 2013: 349381
Simultaneous embedding is concerned with simultaneously representing a series of graphs sharing some or all vertices. This forms the basis for the visualization of dynamic graphs and thus is an important field of research. Recently there has been a great deal of work investigating simultaneous embedding problems both from a theoretical and a practical point of view. We survey recent work on this topic.