Strip Planarity Testing for Embedded Planar Graphs

Angelini, Patrizio; Lozzo, Giordano Da; Battista, Giuseppe Di; Frati, Fabrizio

doi:10.1007/s00453-016-0128-9

Cited by 21 publications

(31 citation statements)

References 46 publications

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“…A great body of literature is devoted to the study of constrained notions of planarity. Classical examples are clustered planarity [3,7,14], in which vertices are constrained into prescribed regions of the plane called clusters, level planarity [4,19], in which vertices are assigned to horizontal lines, strip planarity [2], in which vertices have to lie inside parallel strips of the plane, and upward planarity. A directed acyclic graph is upward-planar if it admits a planar drawing in which, for each directed edge (u, v), vertex u lies below v and (u, v) is represented by a y-monotone curve.…”

Section: Introductionmentioning

confidence: 99%

Windrose Planarity: Embedding Graphs with Direction-Constrained Edges

Angelini¹,

Lozzo²,

Battista³

et al. 2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

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Given a planar graph G and a partition of the neighbors of each vertex v in four sets v, v, v, and v, the problem Windrose Planarity asks to decide whether G admits a windrose-planar drawing, that is, a planar drawing in which (i) each neighbor u ∈ v is above and to the right of v, (ii) each neighbor u ∈ v is above and to the left of v, (iii) each neighbor u ∈ v is below and to the left of v, (iv) each neighbor u ∈ v is below and to the right of v, and (v) edges are represented by curves that are monotone with respect to each axis. By exploiting both the horizontal and the vertical relationship among vertices, windrose-planar drawings allow to simultaneously visualize two partial orders defined by means of the edges of the graph.Although the problem is N P-hard in the general case, we give a polynomial-time algorithm for testing whether there exists a windrose-planar drawing that respects a given combinatorial embedding. This algorithm is based on a characterization of the plane triangulations admitting a windrose-planar drawing. Furthermore, for any embedded graph with n vertices that has a windrose-planar drawing, we can construct one with at most one bend per edge and with at most 2n − 5 bends in total, which lies on the 3n × 3n grid. The latter result contrasts with the fact that straight-line windrose-planar drawings may require exponential area.

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Section: Introductionmentioning

confidence: 99%

Windrose Planarity: Embedding Graphs with Direction-Constrained Edges

Angelini¹,

Lozzo²,

Battista³

et al. 2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

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“…FPT algorithms have also been investigated [10,15]. For additional special cases, see, e.g., [2,3,4,7,14,23].A c-graph is flat when no non-trivial cluster is a subset of another, so T has only three levels: the root, the clusters, and the leaves. Flat C-Planarity can be solved in polynomial time for embedded c-graphs with at most 5 vertices per face [22,26] or at most two vertices of each cluster per face [13], for embedded c-graphs in which each cluster induces a subgraph with at most two connected components [30], and for c-graphs with two clusters [9,26,29] or three clusters [1].…”

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confidence: 99%

Subexponential-Time and FPT Algorithms for Embedded Flat Clustered Planarity

Lozzo

Eppstein

Goodrich

et al. 2018

Graph-Theoretic Concepts in Computer Science

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The C-Planarity problem asks for a drawing of a clustered graph, i.e., a graph whose vertices belong to properly nested clusters, in which each cluster is represented by a simple closed region with no edgeedge crossings, no region-region crossings, and no unnecessary edge-region crossings. We study C-Planarity for embedded flat clustered graphs, graphs with a fixed combinatorial embedding whose clusters partition the vertex set. Our main result is a subexponential-time algorithm to test C-Planarity for these graphs when their face size is bounded. Furthermore, we consider a variation of the notion of embedded tree decomposition in which, for each face, including the outer face, there is a bag that contains every vertex of the face. We show that C-Planarity is fixed-parameter tractable with the embedded-width of the underlying graph and the number of disconnected clusters as parameters.A clustered graph (or c-graph) is a pair C (G, T ) with underlying graph G and inclusion tree T , i.e., a rooted tree whose leaves are the vertices of G. Each internal node µ of T represents a cluster of vertices of G (its leaf descendants) which induces a subgraph G(µ). A c-planar drawing of C (G, T ) ( Fig. 1) consists of a drawing of G and of a representation of each cluster µ as a simple closed region R(µ), i.e., a region homeomorphic to a closed disc, such that: (1) Each region R(µ) contains the drawing of G(µ). (2) For every two clusters µ, ν ∈ T , R(ν) ⊆ R(µ) if and only if ν is a descendant of µ in T . (3) No two edges cross. (4) No edge crosses any region boundary more than once. (5) No two region boundaries intersect.An interesting and challenging line of research in graph drawing concerns the computational complexity of the C-Planarity problem, which asks to test the existence of a c-planar drawing of a c-graph. This problem is notoriously difficult, particularly when (as in Fig. 1) clusters may be disconnected, faces may have unbounded size, and the cluster hierarchy may have multiple nested levels. No known subexponential-time algorithm solves the (general) C-Planarity problem, and it is unknown whether it is NP-complete, although the related problem of splitting as few clusters as possible to make a c-graph c-planar was proved NP-hard [5]. Thus, there is considerable interest in subexponential-time, slice-wise polynomial, and fixed-parameter tractable algorithms, besides polynomial-time algorithms for special cases of C-Planarity.C-Planarity was introduced by Feng, Cohen, and Eades [24], who solved it in quadratic time for the c-connected case when every cluster induces a connected subgraph. Similar results were given by Lengauer [32] using different terminology. Dahlhaus [21] claimed a linear-time algorithm for c-connected C-Planarity (with some details later provided by Cortese et al. [18]). Goodrich et al. [27] gave a cubic-time algorithm for disconnected clusters that satisfy an "extroverted" property, and Gutwenger et al.[28] provided a polynomial-time algorithm for "almost" c-connected inputs. Cornelsen and Wagne...

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“…The goal is to find a planar drawing of G in which vertices lie inside the corresponding strips and edges cross the boundary of any strip at most once. We observe that STRIP PLANARITY is equivalent to C-PLANARITY WITH EMBEDDED PIPES when G A is a path [2].…”

Section: Single-source Strip Planaritymentioning

confidence: 82%

Clustered Planarity with Pipes

Angelini

Lozzo

2019

Algorithmica

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We study the version of the C-PLANARITY problem in which edges connecting the same pair of clusters must be grouped into pipes, which generalizes the STRIP PLANARITY problem. We give algorithms to decide several families of instances for the two variants in which the order of the pipes around each cluster is given as part of the input or can be chosen by the algorithm. IntroductionVisualizing clustered graphs is a challenging task with several applications in the analysis of networks that exhibit a hierarchical structure. The most established criterion for a readable visualization of these graphs has been formalized in the notion of c-planarity, introduced by Feng, Cohen, and Eades [13] in 1995. Given a clustered graph C (G, T ) (c-graph), that is, a graph G equipped with a recursive clustering T of its vertices, the C-PLANARITY problem asks whether there exist a planar drawing of G and a representation of each cluster as a topological disk enclosing all and only its vertices, such that no "unnecessary" crossings occur between disks and edges, or between disks. Ever since its introduction, this problem has been attracting a great deal of research. However, the question regarding its computational complexity withstood the attack of several powerful algorithmic tools, such as the Hanani-Tutte theorem [14,16], the SPQR-tree machinery [10], and the Simultaneous PQ-ordering framework [6].The clustering of a c-graph C (G, T ) is described by a rooted tree T whose leaves are the vertices of G and whose each internal node µ different from the root represents a cluster containing all and only the leaves of the subtree of T rooted at µ. A c-graph is flat if T has height 2. The clusters-adjacency graph G A of a flat c-graph is the graph obtained from the c-graph by contracting each cluster into a single vertex, and by removing multi-edges and loops.Cortese et al.[11] introduced a variant of C-PLANARITY for flat c-graphs, which we call C-PLANARITY WITH EMBEDDED PIPES. The input of this problem is a flat c-graph C (G, T ) together with a planar drawing of its clusters-adjacency graph G A , in which vertices of G A are represented by disks and edges of G A by pipes connecting the disks. The goal is then to produce a c-planar drawing of C (G, T ) in which each vertex of G lies inside the disk representing the cluster it belongs to and each inter-cluster edge of G is drawn inside the corresponding pipe. In [11] this problem is solved when the underlying graph G is a cycle. Chang, Erickson, and Xu [9] observed that in this case the problem is equivalent to determining whether a closed walk of length n in a simple plane graph is weakly simple, and improved the time complexity to O(n log n). The special case of the problem in which the clusters-adjacency graph is a path while G can be any planar graph, which is known by the name of STRIP PLANARITY, has also been studied. Polynomial-time algorithms for this problem have been presented when the underlying graph has a fixed planar embedding [2] and when it is a tree [14].We remark th...

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Strip Planarity Testing for Embedded Planar Graphs

Cited by 21 publications

References 46 publications

Windrose Planarity: Embedding Graphs with Direction-Constrained Edges

Windrose Planarity: Embedding Graphs with Direction-Constrained Edges

Subexponential-Time and FPT Algorithms for Embedded Flat Clustered Planarity

Clustered Planarity with Pipes

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