Advances on Testing C-Planarity of Embedded Flat Clustered Graphs

Chimani, Markus; Battista, Giuseppe Di; Frati, Fabrizio; Klein, Karsten

doi:10.1007/978-3-662-45803-7_35

Cited by 12 publications

(6 citation statements)

References 25 publications

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“…The result also follows from a work by Gutwenger et al [17]. Beyond two clusters a polynomial time algorithm for c-planarity was obtained only in special cases, e.g., [8,16,17,20,21], and most recently in [6,7]. Cortese et al [9] shows that c-planarity is solvable in…”

Section: Introductionsupporting

confidence: 56%

“…In the general case (including already the case of three clusters) of so-called flat clustered graphs a similar result was obtained only in very limited cases. Specifically, either when every face of G is incident to at most five vertices [10,15], or when there exist at most two vertices of a cluster incident to a single face [7]. We remark that the techniques of the previously mentioned papers do not give a polynomial-time algorithm for the case of three clusters, and also do not seem to be adaptable to this setting.…”

Section: Introductionmentioning

confidence: 97%

“…On the other hand, once we have a clustered embedding of G we can augment G by adding edges drawn inside clusters without creating an edge-crossing so that clusters become connected. These observations suggest that c-planarity of G could be viewed as a connectivity augmentation problem, for example as in [7,15], in which we want to decide if it is possible to make clusters connected while maintaining the planarity of G. One minor problem with this viewpoint is the fact that if G is c-planar we do not allow a cluster V i to induce a cycle such that clusters V j and V j , i = j, j , are drawn on its opposite sides. However, this cannot happen if G is cyclic.…”

Section: Introductionmentioning

confidence: 99%

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C-Planarity of Embedded Cyclic c-Graphs

Fulek

2016

Lecture Notes in Computer Science

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Abstract. We show that c-planarity is solvable in quadratic time for flat clustered graphs with three clusters if the combinatorial embedding of the underlying graph is fixed. In simpler graph-theoretical terms our result can be viewed as follows. Given a graph G with the vertex set partitioned into three parts embedded on a 2-sphere, our algorithm decides if we can augment G by adding edges without creating an edge-crossing so that in the resulting spherical graph the vertices of each part induce a connected sub-graph. We proceed by a reduction to the problem of testing the existence of a perfect matching in planar bipartite graphs. We formulate our result in a slightly more general setting of cyclic clustered graphs, i.e., the simple graph obtained by contracting each cluster, where we disregard loops and multi-edges, is a cycle.

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Section: Introductionsupporting

confidence: 56%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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C-Planarity of Embedded Cyclic c-Graphs

Fulek

2016

Lecture Notes in Computer Science

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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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“…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]. At the other end of the size spectrum, Jelínková et al[31] provide efficient algorithms for 3-connected flat c-graphs when each cluster has at most 3 vertices.…”

mentioning

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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Advances on Testing C-Planarity of Embedded Flat Clustered Graphs

Abstract: We show a polynomial-time algorithm for testing c-planarity of embedded flat clustered graphs with at most two vertices per cluster on each face.

Cited by 12 publications

References 25 publications

C-Planarity of Embedded Cyclic c-Graphs

C-Planarity of Embedded Cyclic c-Graphs

Windrose Planarity: Embedding Graphs with Direction-Constrained Edges

Subexponential-Time and FPT Algorithms for Embedded Flat Clustered Planarity

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