Clustered Planarity: Embedded Clustered Graphs with Two-Component Clusters

Jelínek, Vít; Jelínková, Eva; Kratochvı́l, Jan; Lidický, Bernard

doi:10.1007/978-3-642-00219-9_13

Cited by 21 publications

(17 citation statements)

References 12 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: 57%

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: 57%

C-Planarity of Embedded Cyclic c-Graphs

Fulek

2016

Lecture Notes in Computer Science

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“…That is, every cluster is a child of the root of T and a combinatorial embedding for G (i.e., an order of the edges incident to each vertex) is fixed in advance; then, the c-planarity testing problem asks whether a c-planar drawing exists in which G has the prescribed combinatorial embedding. This natural variant of the c-planarity testing problem is well-studied [11,12,14,28,30], due to the fact that several NP-hard graph drawing problems are polynomial-time solvable in the fixed embedding scenario [7,23,36] and that testing c-planarity of embedded flat clustered graphs generalizes testing c-planarity of the notable class of triconnected flat clustered graphs. Yet determining the time complexity of testing c-planarity for this innocent-looking case eludes an answer.…”

Section: Strip Planarity and Clustered Planaritymentioning

confidence: 99%

Strip Planarity Testing for Embedded Planar Graphs

et al. 2016

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In this paper we introduce and study the strip planarity testing problem, which takes as an input a planar graph G(V, E) and a function γ : V → {1, 2, . . . , k} and asks whether a planar drawing of G exists such that each edge is represented by a curve that is monotone in the y-direction and, for any u, v ∈ V with γ (u) < γ (v), it holds that y(u) < y(v). The problem has strong relationships with some of the most deeply studied variants of the planarity testing problem, such as clustered planarity, upward planarity, and level planarity. Most notably, we provide a polynomial-time reduction from strip planarity testing to clustered planarity. We show that the strip planarity testing problem is polynomial-time solvable if G has a prescribed combinatorial embedding.

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“…Currently only some special graph classes are known to be solvable in polynomial time, e.g., [4,7,8,[10][11][12]. Most importantly, there is a linear-time algorithm to check cluster-connected graphs (i.e., G [σ] is connected for each σ) [5].…”

Section: Introductionmentioning

confidence: 99%

Shrinking the Search Space for Clustered Planarity

Chimani

Klein

2013

Graph Drawing

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Abstract.A clustered graph is a graph augmented with a hierarchical inclusion structure over its vertices, and arises very naturally in multiple application areas. While it is long known that planarity-i.e., drawability without edge crossingsof graphs can be tested in polynomial (linear) time, the complexity for the clustered case is still unknown.In this paper, we present a new graph theoretic reduction which allows us to considerably shrink the combinatorial search space, which is of benefit for all enumeration-type algorithms. Based thereon, we give new classes of polynomially testable graphs and a practically efficient exact planarity test for general clustered graphs based on an integer linear program.

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Clustered Planarity: Embedded Clustered Graphs with Two-Component Clusters

Abstract: We present a polynomial-time algorithm for c-planarity testing of clustered graphs with fixed plane embedding and such that every cluster induces a subgraph with at most two connected components.

Cited by 21 publications

References 12 publications

C-Planarity of Embedded Cyclic c-Graphs

C-Planarity of Embedded Cyclic c-Graphs

Strip Planarity Testing for Embedded Planar Graphs

Shrinking the Search Space for Clustered Planarity

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