Clustered Planarity Testing Revisited

Fulek, Radoslav; Kynčl, Jan; Malinović, Igor; Pálvölgyi, Dömötör

doi:10.1007/978-3-662-45803-7_36

Cited by 15 publications

(40 citation statements)

References 41 publications

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“…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%

“…. and V c−1 , and a given combinatorial embedding, and we assume that G is cyclic [15,Section 6]. Thus, every e = uv of G is such that u ∈ V i and v ∈ V j where j − i mod c ≤ 1 and for every i there exists an edge in G between V i and V i+1 mod c .…”

Section: Introductionmentioning

confidence: 99%

“…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: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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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“…Hanani-Tutte style characterizations have also been established for partially embedded planar graphs, several classes of simultaneously embedded planar graphs [18], and two-clustered graphs [8]. The family of counterexamples in [8,Section 4] shows that a straightforward variant of the Hanani-Tutte theorem for clustered graphs with more than two clusters fails.…”

Section: Introductionmentioning

confidence: 99%

“…The family of counterexamples in [8,Section 4] shows that a straightforward variant of the Hanani-Tutte theorem for clustered graphs with more than two clusters fails. Gutwenger et al [11] showed that by using the reduction from [18], this counterexample can be turned into a counterexample for a variant of the Hanani-Tutte theorem for two simultaneously embedded planar graphs [18,Conjecture 6.20].…”

Section: Introductionmentioning

confidence: 99%

Hanani-Tutte for Radial Planarity

Fulek¹,

Pelsmajer²,

Schaefer³

2017

JGAA

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Abstract. A drawing of a graph G is radial if the vertices of G are placed on concentric circles C1, . . . , C k with common center c, and edges are drawn radially: every edge intersects every circle centered at c at most once. G is radial planar if it has a radial embedding, that is, a crossing-free radial drawing. If the vertices of G are ordered or partitioned into ordered levels (as they are for leveled graphs), we require that the assignment of vertices to circles corresponds to the given ordering or leveling. We show that a graph G is radial planar if G has a radial drawing in which every two edges cross an even number of times; the radial embedding has the same leveling as the radial drawing. In other words, we establish the weak variant of the Hanani-Tutte theorem for radial planarity. This generalizes a result by Pach and Tóth.

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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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Clustered Planarity Testing Revisited

Cited by 15 publications

References 41 publications

C-Planarity of Embedded Cyclic c-Graphs

C-Planarity of Embedded Cyclic c-Graphs

Hanani-Tutte for Radial Planarity

Subexponential-Time and FPT Algorithms for Embedded Flat Clustered Planarity

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