C-Planarity of Embedded Cyclic c-Graphs

Fulek, Radoslav

doi:10.1007/978-3-319-50106-2_8

Cited by 3 publications

(2 citation statements)

References 25 publications

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“…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. Fulek [25] speculates that C-Planarity could be solvable in subexponential time for more general embedded flat c-graphs.…”

Section: Introductionmentioning

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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Section: Introductionmentioning

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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“…Though, in [2] related SPQR-trees were employed to improve upon our running time. A recent work [3,18] suggested an approach via computation of a flow/perfect matching in a graph. However, this approach is tailored for the setting in which also the isotopy class of an embedding of G is fixed.…”

Section: The Resultsmentioning

confidence: 99%

Hanani-Tutte for approximating maps of graphs

Fulek¹,

Kynčl²

2017

Preprint

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We resolve in the affirmative conjectures of Repovš andA. Skopenkov (1998), andM. Skopenkov (2003) generalizing the classical Hanani-Tutte theorem to the setting of approximating maps of graphs on 2-dimensional surfaces by embeddings. Our proof of this result is constructive and almost immediately implies an efficient algorithm for testing whether a given piecewise linear map of a graph in a surface is approximable by an embedding. More precisely, an instance of this problem consists of (i) a graph G whose vertices are partitioned into clusters and whose inter-cluster edges are partitioned into bundles, and (ii) a region R of a 2-dimensional compact surface M given as the union of a set of pairwise disjoint discs corresponding to the clusters and a set of pairwise disjoint "pipes" corresponding to the bundles, connecting certain pairs of these discs. We are to decide whether G can be embedded inside M so that the vertices in every cluster are drawn in the corresponding disc, the edges in every bundle pass only through its corresponding pipe, and every edge crosses the boundary of each disc at most once.

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Bounded Embeddings of Graphs in the Plane

Fulek

2016

Lecture Notes in Computer Science

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A drawing in the plane (R 2 ) of a graph G = (V, E) equipped with a function γ :, where x(.) denotes the projection to the x-axis. We prove a characterization of isotopy classes of graph embeddings in the plane containing an x-bounded embedding.Then we present an efficient algorithm, that relies on our result, for testing the existence of an x-bounded embedding if the given graph is a tree or generalized Θ-graph. This partialy answers a question raised recently by Angelini et al. and Chang et al., and proves that c-planarity testing of flat clustered graphs with three clusters is tractable if each connected component of the underlying abstract graph is a tree. * The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007(FP7/ -2013 under REA grant agreement no [291734].

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C-Planarity of Embedded Cyclic c-Graphs

Cited by 3 publications

References 25 publications

Subexponential-Time and FPT Algorithms for Embedded Flat Clustered Planarity

Subexponential-Time and FPT Algorithms for Embedded Flat Clustered Planarity

Hanani-Tutte for approximating maps of graphs

Bounded Embeddings of Graphs in the Plane

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