On Embedding a Cycle in a Plane Graph

Cortese, Pier Francesco; Battista, Giuseppe Di; Patrignani, Maurizio; Pizzonia, Maurizio

doi:10.1007/11618058_5

Cited by 10 publications

(6 citation statements)

References 12 publications

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“…However, we will prove that a different reduction from (G, γ ) yields a flat clustered graph C(G, T ) whose cplanarity is in fact a necessary and sufficient condition for the strip planarity of (G, γ ); in other words, we will prove that the strip planarity testing problem reduces in polynomial time to the c-planarity testing problem for flat clustered graphs. Furthermore, it turns out that strip planarity testing coincides with a special case of a problem posed by Cortese et al [12,13] and related to c-planarity testing. The problem asks whether a graph G can be planarly embedded "inside" an host graph H , which can be thought as having "fat" vertices and edges, with each vertex and edge of G drawn inside a prescribed vertex and a prescribed edge of H , respectively.…”

Section: Strip Planarity and Clustered Planaritymentioning

confidence: 90%

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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Section: Strip Planarity and Clustered Planaritymentioning

confidence: 90%

Strip Planarity Testing for Embedded Planar Graphs

et al. 2016

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“…1(a)). It turns out that strip planarity testing coincides with a special case of a problem opened by Cortese et al [8,9] and related to c-planarity testing. The problem asks whether a graph G can be planarly embedded "inside" an host graph H, which can be thought as having "fat" vertices and edges, with each vertex and edge of G drawn inside a prescribed vertex and a prescribed edge of H, respectively.…”

Section: Introductionmentioning

confidence: 76%

Strip Planarity Testing

et al. 2013

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Abstract. 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 monotone in the y-direction and, for any u, v ∈ V with γ(u) < γ(v), it holds 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. We show that the problem is polynomial-time solvable if G has a fixed planar embedding.

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“…• Cortese et al showed that the c-planarity testing and embedding problem is polynomial-time solvable for rigid clustered cycles, that is, for flat clustered graphs where the underlying graph is a cycle and the graph of the clusters' adjacencies is planar and has a fixed embedding [8].…”

Section: Introductionmentioning

confidence: 99%

C-Planarity of C-Connected Clustered Graphs

Cortese¹,

Battista²,

Frati³

et al. 2008

JGAA

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We present the first characterization of c-planarity for c-connected clustered graphs. The characterization is based on the interplay between the hierarchy of the clusters and the hierarchies of the triconnected and biconnected components of the underlying graph. Based on such a characterization, we provide a linear-time c-planarity testing and embedding algorithm for c-connected clustered graphs. The algorithm is reasonably easy to implement, since it exploits as building blocks simple algorithmic tools like the computation of lowest common ancestors, minimum and maximum spanning trees, and counting sorts. It also makes use of well-known data structures as SPQR-trees and BC-trees. If the test fails, the algorithm identifies a structural element responsible for the non-cplanarity of the input clustered graph.

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On Embedding a Cycle in a Plane Graph

Cited by 10 publications

References 12 publications

Strip Planarity Testing for Embedded Planar Graphs

Strip Planarity Testing for Embedded Planar Graphs

Strip Planarity Testing

C-Planarity of C-Connected Clustered Graphs

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