Efficient C-Planarity Testing for Embedded Flat Clustered Graphs with Small Faces

Battista, Giuseppe Di; Frati, Fabrizio

doi:10.7155/jgaa.00191

Cited by 33 publications

(36 citation statements)

References 13 publications

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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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Section: Strip Planarity and Clustered Planaritymentioning

confidence: 99%

Strip Planarity Testing for Embedded Planar Graphs

et al. 2016

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“…Di Battista and Frati [22] gave the following characterization. (i) G is planar; (ii) there exists a face f in G such that when f is chosen as the outer face for G no cycle composed of vertices of the same cluster encloses a vertex of a different cluster in its interior; and (iii) there exists a super c-graph C (G , T ) of C such that G is planar and C is c-connected (see Fig.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, we can assume that any c-graph satisfies these conditions. Following [22] we thus view the problem of testing c-planarity as one of testing Condition (iii).…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

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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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“…However, one is often not interested in just any planar embedding, but in one that has some additional properties. Examples of such properties include that a given existing partial drawing should be extended [3,16] or that some parts of the graph should appear clustered together [10,17].…”

Section: Introductionmentioning

confidence: 99%

Multilevel Planarity

Barth¹,

Brückner²,

Jungeblut³

et al. 2018

WALCOM: Algorithms and Computation

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In this paper, we introduce and study the multilevel-planarity testing problem, which is a generalization of upward planarity and level planarity. Let G = (V, E) be a directed graph and let : V → P(Z) be a function that assigns a finite set of integers to each vertex. A multilevelplanar drawing of G is a planar drawing of G such that the y-coordinate of each vertex v ∈ V is y(v) ∈ (v), and each edge is drawn as a strictly ymonotone curve. We present linear-time algorithms for testing multilevel planarity of embedded graphs with a single source and of oriented cycles. Complementing these algorithmic results, we show that multilevel-planarity testing is NPcomplete even in very restricted cases.

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Efficient C-Planarity Testing for Embedded Flat Clustered Graphs with Small Faces

Cited by 33 publications

References 13 publications

Strip Planarity Testing for Embedded Planar Graphs

Strip Planarity Testing for Embedded Planar Graphs

Subexponential-Time and FPT Algorithms for Embedded Flat Clustered Planarity

Multilevel Planarity

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