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**11 - 14**of**14**### Simultaneous Embeddability of Two Partitions

"... Abstract. We study the simultaneous embeddability of a pair of partitions of the same underlying set into disjoint blocks. Each element of the set is mapped to a point in the plane and each block of either of the two partitions is mapped to a region that contains exactly those points that belong to ..."

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Abstract. We study the simultaneous embeddability of a pair of partitions of the same underlying set into disjoint blocks. Each element of the set is mapped to a point in the plane and each block of either of the two partitions is mapped to a region that contains exactly those points that belong to the elements in the block and that is bounded by a simple closed curve. We establish three main classes of simultaneous embeddability (weak, strong, and full embeddability) that differ by increasingly strict well-formedness conditions on how different block regions are allowed to intersect. We show that these simultaneous embeddability classes are closely related to different planarity concepts of hypergraphs. For each embeddability class we give a full characterization. We show that (i) every pair of partitions has a weak simultaneous embedding, (ii) it is NP-complete to decide the existence of a strong simultaneous embedding, and (iii) the existence of a full simultaneous embedding can be tested in linear time. 1

### On the Complexity of Clustered-Level Planarity and T-Level Planarity?

"... Abstract. In this paper we study two problems related to the drawing of level graphs, that is, T-LEVEL PLA-NARITY and CLUSTERED-LEVEL PLANARITY. We show that both problems are NP-complete in the general case and that they become polynomial-time solvable when restricted to proper instances. 1 ..."

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Abstract. In this paper we study two problems related to the drawing of level graphs, that is, T-LEVEL PLA-NARITY and CLUSTERED-LEVEL PLANARITY. We show that both problems are NP-complete in the general case and that they become polynomial-time solvable when restricted to proper instances. 1

### Advancements on SEFE and Partitioned Book Embedding Problems

"... Abstract. In this work we investigate the complexity of some problems related to the Simultaneous Embedding with Fixed Edges (SEFE) of k planar graphs and the PARTITIONED k-PAGE BOOK EMBEDDING (PBE-k) problems, which are known to be equivalent under certain conditions. While the computational comple ..."

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Abstract. In this work we investigate the complexity of some problems related to the Simultaneous Embedding with Fixed Edges (SEFE) of k planar graphs and the PARTITIONED k-PAGE BOOK EMBEDDING (PBE-k) problems, which are known to be equivalent under certain conditions. While the computational complexity of SEFE for k = 2 is still a central open question in Graph Drawing, the problem is NP-complete for k ≥ 3 [Gassner et al., WG ’06], even if the intersection graph is the same for each pair of graphs (sunflower intersection) [Schaefer, JGAA (2013)]. We improve on these results by proving that SEFE with k ≥ 3 and sunflower intersection is NP-complete even when the intersection graph is a tree and all the input graphs are biconnected. Also, we prove NP-completeness for k ≥ 3 of problem PBE-k and of problem PARTITIONED T-COHERENT k-PAGE BOOK EMBEDDING (PTBE-k)- that is the generalization of PBE-k in which the order-ing of the vertices on the spine is constrained by a tree T- even when two input graphs are biconnected. Further, we provide a linear-time algorithm for PTBE-k when k − 1 pages are assigned a connected graph. Finally, we prove that the problem of maximizing the number of edges that are drawn the same in a SEFE of two graphs isNP-complete in several restricted settings (optimization version

### Drawings of Non-planar Graphs with Crossing-free Subgraphs?

"... Abstract. We initiate the study of the following problem: Given a non-planar graph G and a planar subgraph S of G, does there exist a straight-line drawing Γ of G in the plane such that the edges of S are not crossed in Γ? We give positive and negative results for different kinds of spanning subgrap ..."

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Abstract. We initiate the study of the following problem: Given a non-planar graph G and a planar subgraph S of G, does there exist a straight-line drawing Γ of G in the plane such that the edges of S are not crossed in Γ? We give positive and negative results for different kinds of spanning subgraphs S of G. Moreover, in order to enlarge the subset of instances that admit a solution, we consider the possibility of bending the edges of G \ S; in this setting different trade-offs between number of bends and drawing area are given. 1