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14
Untangling two systems of noncrossing curves
"... 2014 We consider two systems (α1,..., αm) and (β1,..., βn) of curves drawn on a compact twodimensional surface M with boundary. Each αi and each βj is either an arc meeting the boundary of M at its two endpoints, or a closed curve. The αi are pairwise disjoint except for possibly sharing endpoints, ..."
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2014 We consider two systems (α1,..., αm) and (β1,..., βn) of curves drawn on a compact twodimensional surface M with boundary. Each αi and each βj is either an arc meeting the boundary of M at its two endpoints, or a closed curve. The αi are pairwise disjoint except for possibly sharing endpoints, and similarly for the βj. We want to “untangle ” the βj from the αi by a selfhomeomorphism of M; more precisely, we seek an homeomorphism ϕ: M → M fixing the boundary of M pointwise such that the total number of crossings of the αi with the ϕ(βj) is as small as possible. This problem is motivated by an application in the algorithmic theory of embeddings and 3manifolds. We prove that if M is planar, i.e., a sphere with h ≥ 0 boundary components (“holes”), then O(mn) crossings can be achieved (independently of h), which is asymptotically tight, as an easy lower bound shows. In general, for an arbitrary (orientable or nonorientable) surface M with h holes and of (orientable or nonorientable) genus g ≥ 0, we obtain an O((m + n)4) upper bound, again independent of h and g. The proofs rely, among other things, on a result concerning simultaneous planar drawings of graphs by Erten and Kobourov. 1
The complexity of simultaneous geometric graph embedding
, 2015
"... Given a collection of planar graphs G1,..., Gk on the same set V of n vertices, the simultaneous geometric embedding (with mapping) problem, or simply kSGE, is to find a set P of n points in the plane and a bijection ϕ: V → P such that the induced straightline drawings of G1,..., Gk under ϕ are al ..."
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Given a collection of planar graphs G1,..., Gk on the same set V of n vertices, the simultaneous geometric embedding (with mapping) problem, or simply kSGE, is to find a set P of n points in the plane and a bijection ϕ: V → P such that the induced straightline drawings of G1,..., Gk under ϕ are all plane. This problem is polynomialtime equivalent to weak rectilinear realizability of abstract topological graphs, which Kynčl (doi:10.1007/s004540109320x) proved to be complete for ∃R, the existential theory of the reals. Hence the problem kSGE is polynomialtime equivalent to several other problems in computational geometry, such as recognizing intersection graphs of line segments or finding the rectilinear crossing number of a graph. We give an elementary reduction from the pseudoline stretchability problem to kSGE, with the property that both numbers k and n are linear in the number of pseudolines. This implies not only the ∃Rhardness result, but also a 22 Ω(n) lower bound on the minimum size of a grid on which any such simultaneous embedding can be drawn. This bound is tight. Hence there exists such collections of graphs that can be simultaneously embedded, but every simultaneous drawing requires an exponential number of bits per coordinates. The best value that can be extracted from Kynčl’s proof is only 22 Ω(
Column planarity and partial simultaneous geometric embedding
 In Proc. 22nd International Symposium on Graph Drawing (GD2014
, 2014
"... Abstract. We introduce the notion of column planarity of a subset R of the vertices of a graph G. Informally, we say that R is column planar in G if we can assign xcoordinates to the vertices in R such that any assignment of ycoordinates to them produces a partial embedding that can be completed t ..."
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Abstract. We introduce the notion of column planarity of a subset R of the vertices of a graph G. Informally, we say that R is column planar in G if we can assign xcoordinates to the vertices in R such that any assignment of ycoordinates to them produces a partial embedding that can be completed to a plane straightline drawing of G. Column planarity is both a relaxation and a strengthening of unlabeled level planarity. We prove near tight bounds for column planar subsets of trees: any tree on n vertices contains a column planar set of size at least 14n/17 and for any > 0 and any sufficiently large n, there exists an nvertex tree in which every column planar subset has size at most (5/6 + )n. We also consider a relaxation of simultaneous geometric embedding (SGE), which we call partial SGE (PSGE). A PSGE of two graphs G1 and G2 allows some of their vertices to map to two different points in the plane. We show how to use column planar subsets to construct kPSGEs in which k vertices are still mapped to the same point. In particular, we show that any two trees on n vertices admit an 11n/17PSGE, two outerpaths admit an n/4PSGE, and an outerpath and a tree admit a 11n/34PSGE. 1
Drawing partially embedded and simultaneously planar graphs
 22nd International Symposium on Graph Drawing (GD ’14
"... Abstract. We investigate the problem of constructing planar drawings with few bends for two related problems, the partially embedded graph problem—to extend a straightline planar drawing of a subgraph to a planar drawing of the whole graph—and the simultaneous planarity problem—to find planar draw ..."
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Abstract. We investigate the problem of constructing planar drawings with few bends for two related problems, the partially embedded graph problem—to extend a straightline planar drawing of a subgraph to a planar drawing of the whole graph—and the simultaneous planarity problem—to find planar drawings of two graphs that coincide on shared vertices and edges. In both cases we show that if the required planar drawings exist, then there are planar drawings with a linear number of bends per edge and, in the case of simultaneous planarity, a constant number of crossings between every pair of edges. Our proofs provide efficient algorithms if the combinatorial embedding of the drawing is given. Our result on partially embedded graph drawing generalizes a classic result by Pach and Wenger which shows that any planar graph can be drawn with a linear number of bends per edge if the location of each vertex is fixed. 1
DOI: 10.7155/jgaa.00375 Drawing Partially Embedded and Simultaneously Planar Graphs
, 2015
"... We investigate the problem of constructing planar drawings with few bends for two related problems, the partially embedded graph problem—to extend a straightline planar drawing of a subgraph to a planar drawing of the whole graph—and the simultaneous planarity problem—to find planar drawings of two ..."
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We investigate the problem of constructing planar drawings with few bends for two related problems, the partially embedded graph problem—to extend a straightline planar drawing of a subgraph to a planar drawing of the whole graph—and the simultaneous planarity problem—to find planar drawings of two graphs that coincide on shared vertices and edges. In both cases we show that if the required planar drawings exist, then there are planar drawings with a linear number of bends per edge and, in the case of simultaneous planarity, with a number of crossings between any pair of edges which is bounded by a constant. Our proofs provide efficient algorithms if the combinatorial embedding of the drawing is given. Our result on partially embedded graph drawing generalizes a classic result by Pach and Wenger which shows that any planar graph can be drawn with a linear number of bends per edge if the location of each vertex is fixed. Submitted:
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"... A summary graph (middle) constructed from four graphs of molecules; colors indicate which input graphs contain a given feature with gray indicating the feature is common to all graphs. Graphs can be used to represent a variety of information, from molecular structures to biological pathways to compu ..."
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A summary graph (middle) constructed from four graphs of molecules; colors indicate which input graphs contain a given feature with gray indicating the feature is common to all graphs. Graphs can be used to represent a variety of information, from molecular structures to biological pathways to computational workflows. With a growing volume of data represented as graphs, the problem of understanding and analyzing the variations in a collection of graphs is of increasing importance. We present an algorithm to compute a single summary graph that efficiently encodes an entire collection of graphs by finding and merging similar nodes and edges. Instead of only merging nodes and edges that are exactly the same, we use domainspecific comparison functions to collapse similar nodes and edges which allows us to generate more compact representations of the collection. In addition, we have developed methods that allow users to interactively control the display of these summary graphs. These interactions include the ability to highlight individual graphs in the summary, control the succinctness of the summary, and explicitly define when specific nodes should or should not be merged. We show that our approach to generating and interacting with graph summaries leads to a better understanding of a graph collection by allowing users to more easily identify common substructures and key differences between graphs. 1
SEFE with No Mapping via Large Induced Outerplane Graphs in Plane Graphs
, 2013
"... We show that every nvertex planar graph admits a simultaneous embedding with no mapping and with fixed edges with any (n/2)vertex planar graph. In order to achieve this result, we prove that every nvertex plane graph has an induced outerplane subgraph containing at least n/2 vertices. Also, we ..."
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We show that every nvertex planar graph admits a simultaneous embedding with no mapping and with fixed edges with any (n/2)vertex planar graph. In order to achieve this result, we prove that every nvertex plane graph has an induced outerplane subgraph containing at least n/2 vertices. Also, we show that every nvertex planar graph and every nvertex planar partial 3tree admit a simultaneous embedding with no mapping and with fixed edges.
Monotone Simultaneous Embeddings of Directed Paths∗
, 2014
"... We study monotone simultaneous embeddings of upward planar digraphs, which are simultaneous embeddings where the drawing of each digraph is upward planar, and the directions of the upwardness of different graphs can differ. We first consider the special case where each digraph is a directed path. ..."
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We study monotone simultaneous embeddings of upward planar digraphs, which are simultaneous embeddings where the drawing of each digraph is upward planar, and the directions of the upwardness of different graphs can differ. We first consider the special case where each digraph is a directed path. In contrast to the known result that any two directed paths admit a monotone simultaneous embedding, there exist examples of three paths that do not admit such an embedding for any possible choice of directions of monotonicity. We prove that if a monotone simultaneous embedding of three paths exists then it also exists for any possible choice of directions of monotonicity. We provide a polynomialtime algorithm that, given three paths, decides whether a monotone simultaneous embedding exists and, in the case of existence, also constructs such an embedding. On the other hand, we show that already for three paths, any monotone simultaneous embedding might need a grid of exponential (w.r.t. the number of vertices) size. For more than three paths, we present a polynomialtime algorithm that, given any number of paths and predefined directions of monotonicity, decides whether the paths admit a monotone simultaneous embedding with respect to the given directions, including the construction of a solution if it exists. Further, we show several implications of our results on monotone simultaneous embeddings of general upward planar digraphs. Finally, we discuss complexity issues related to our problems.
Deepening the Relationship between SEFE and CPlanarity
"... Abstract. In this paper we deepen the understanding of the connection between two longstanding Graph Drawing open problems, that is, Simultaneous Embedding with Fixed Edges (SEFE) and Clustered Planarity (CPLANARITY). In ..."
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Abstract. In this paper we deepen the understanding of the connection between two longstanding Graph Drawing open problems, that is, Simultaneous Embedding with Fixed Edges (SEFE) and Clustered Planarity (CPLANARITY). In
Simultaneous Drawing of Planar Graphs with RightAngle Crossings and Few Bends
"... Abstract. Given two planar graphs that are defined on the same set of vertices, a RAC simultaneous drawing is one in which each graph is drawn planar, there are no edge overlaps and the crossings between the two graphs form right angles. The geometric version restricts the problem to straightline d ..."
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Abstract. Given two planar graphs that are defined on the same set of vertices, a RAC simultaneous drawing is one in which each graph is drawn planar, there are no edge overlaps and the crossings between the two graphs form right angles. The geometric version restricts the problem to straightline drawings. It is known, however, that there exists a wheel and a matching which do not admit a geometric RAC simultaneous drawing. In order to enlarge the class of graphs that admit RAC simultaneous drawings, we allow bends in the resulting drawings. We prove that two planar graphs always admit a RAC simultaneous drawing with six bends per edge each, in quadratic area. For more restricted classes of planar graphs (i.e., matchings, paths, cycles, outerplanar graphs and subhamiltonian graphs), we manage to significantly reduce the required number of bends per edge, while keeping the area quadratic. 1