Clustering Cycles into Cycles of Clusters

Discrete Mathematics, 2009

Consider a planar drawing Γ of a planar graph G such that the vertices are drawn as small circles and the edges are drawn as thin stripes. Consider a non-simple cycle c of G. Is it possible to draw c as a non-intersecting closed curve inside Γ , following the circles that correspond in Γ to the vertices of c and the stripes that connect them? We show that this test can be done in polynomial time and study this problem in the framework of clustered planarity for highly non-connected clustered graphs.

2012

In a drawing of a clustered graph vertices and edges are drawn as points and curves, respectively, while clusters are represented by simple closed regions. A drawing of a clustered graph is c-planar if it has no edge-edge, edge-region, or region-region crossings. Determining the complexity of testing whether a clustered graph admits a c-planar drawing is a long-standing open problem in the Graph Drawing research area. An obvious necessary condition for c-planarity is the planarity of the graph underlying the clustered graph. However, such a condition is not sufficient and the consequences on the problem due to the requirement of not having edge-region and region-region crossings are not yet fully understood.

Lecture Notes in Computer Science, 2009

ABSTRACT We present a linear algorithm for c-planarity testing of clustered graphs, in which every cluster has at most four outgoing edges.

Lecture Notes in Computer Science, 2010

In this paper we introduce a generalization of the c-planarity testing problem for clustered graphs. Namely, given a clustered graph, the goal of the SPLIT-C-PLANARITY problem is to split as few clusters as possible in order to make the graph c-planar. Determining whether zero splits are enough coincides with testing c-planarity. We show that SPLIT-C-PLANARITY is NP-complete for c-connected clustered triangulations and for non-c-connected clustered paths and cycles. On the other hand, we present a polynomial-time algorithm for flat c-connected clustered graphs whose underlying graph is a biconnected seriesparallel graph, both in the fixed and in the variable embedding setting, when the splits are assumed to maintain the c-connectivity of the clusters.

Discrete & Computational Geometry, 2011

We show that every c-planar clustered graph has a straight-line c-planar drawing in which each cluster is represented by an axis-parallel rectangle, thus solving a problem posed by Eades, Feng, Lin, and Nagamochi (Algorithmica 44(1):1-32, 2006).

In this paper we study two problems related to the drawing of level graphs, that is, T -LEVEL PLA-NARITY and CLUSTERED-LEVEL PLANARITY. We show that both problems are N P-complete in the general case and that they become polynomial-time solvable when restricted to proper instances.

Algorithmica, 2021

For a clustered graph, i.e, a graph whose vertex set is recursively partitioned into clusters, the C-Planarity Testing problem asks whether it is possible to find a planar embedding of the graph and a representation of each cluster as a region homeomorphic to a closed disk such that (1) the subgraph induced by each cluster is drawn in the interior of the corresponding disk, (2) each edge intersects any disk at most once, and (3) the nesting between clusters is reflected by the representation, i.e., child clusters are properly contained in their parent cluster. The computational complexity of this problem, whose study has been central to the theory of graph visualization since its introduction in 1995 [Feng, Cohen, and Eades, Planarity for clustered graphs, ESA’95], has only been recently settled [Fulek and Tóth, Atomic Embeddability, Clustered Planarity, and Thickenability, to appear at SODA’20]. Before such a breakthrough, the complexity question was still unsolved even when the gr...

Arxiv preprint arXiv: …, 2012

In a drawing of a clustered graph vertices and edges are drawn as points and curves, while clusters are represented by simple closed regions. A drawing is c-planar if it has no edge-edge, edge-region, or region-region crossings. An obvious necessary condition for c-planarity is the planarity of the graph underlying the clustered graph. However, planarity is not sufficient and the constraints imposed by the absence of edge-region and of region-region crossings make the family of c-planar graphs too small for some of the typical Graph Drawing application contexts. Hence, we relax such constraints and define and study α, β, γdrawings of c-graphs whose underlying graph is planar. In an α, β, γdrawing the number of edge-edge, edge-region, and region-region crossings is equal to α, β, and γ, respectively. In this context α, β, γ -drawings are a generalization of c-planar drawings, where α = β = γ = 0.

1997

A graph is a cycle of cliques, if its set of vertices can be partitioned into clusters, such that each cluster is a clique and the cliques form a cycle. Then there is a partition of the set of edges into inner edges of the cliques and interconnection edges between the clusters. Cycles of cliques are a special instance of two-level clustered graphs. Such graphs are drawn by a two phase method: draw the top level graph and then browse into the clusters. In general, it is NP-hard whether or not a graph is a two-level clustered graph of a particular type, e.g. a clique or a planar graph or a triangle of cliques. However, it is efficiently solvable whether or not a graph is a path of cliques or is a large cycle of cliques. * Partially supported by the Deutsche Forschungsgemeinschaft, Forschungsschwerpunkt "Effiziente Algorithmen ffir diskrete Probleme und ihre Anwendungen".

ACM Transactions on Algorithms, 2015

We study the following problem: Given a planar graph G and a planar drawing (embedding) of a subgraph of G, can such a drawing be extended to a planar drawing of the entire graph G? This problem fits the paradigm of extending a partial solution to a complete one, which has been studied before in many different settings. Unlike many cases, in which the presence of a partial solution in the input makes hard an otherwise easy problem, we show that the planarity question remains polynomial-time solvable. Our algorithm is based on several combinatorial lemmata which show that the planarity of partially embedded graphs meets the "oncas" behaviour -obvious necessary conditions for planarity are also sufficient. These conditions are expressed in terms of the interplay between (a) rotation schemes and containment relationships between cycles and (b) the decomposition of a graph into its connected, biconnected, and triconnected components. This implies that no dynamic programming is needed for a decision algorithm and that the elements of the decomposition can be processed independently.

2010

A graph is a (planar, Kh)-graph if a collection of disjoint clusters can be identified such that the subgraph induced by each cluster is an h-clique and collapsing all clusters yields a planar graph. Recognizing (planar, Kh)-graphs is a special instance of the more general problem of recognizing (X ,Y)-graphs, where X and Y are two chosen families of graphs. This model, together with the (X ,Y)-graph terminology, was introduced in [3] and generalized in [1, 2] to support hybrid visualization. In particular, if the clusters are requested to be a partition of the set of the vertices of the input graph G, as it is in [3], then G is called a strong (X ,Y)-graph, otherwise G is a weak (X ,Y)-graph or, simply, an (X ,Y)-graph. In this paper we address (planar, Kh)-recognition in the weak model, and show that this problem is NP-complete if h ≥ 5. This result parallels the analogous result for the strong model [10, 9]. We remark that allowing the contraction of any clique of size greater th...

Journal of Geometry and Physics, 2002

A cluster of cycles (or (r, q)-polycycle) is a simple planar 2-connected finite or countable graph G of girth r and maximal vertex-degree q, which admits an (r, q)-polycyclic realization P (G) on the plane. An (r, q)-polycyclic realization is determined by the following properties: (i) all interior vertices are of degree q, (ii) all interior faces (denote their number by p r ) are combinatorial r-gons, (iii) all vertices, edges and interior faces form a cell-complex.

Combinatorica, 2010

In this paper, we consider the problem for finding an induced cycle passing through k given vertices, which we call the induced cycle problem. The significance of finding induced cycles stems from the fact that precise characterization of perfect graphs would require structures of graphs without an odd induced cycle, and its complement. There has been huge progress in the recent years, especially, the Strong Perfect Graph Conjecture was solved in [6]. Concerning recognition of perfect graphs, there had been a long-standing open problem for detecting an odd hole and its complement, and finally this was solved in [4]. Unfortunately, the problem of finding an induced cycle passing through two given vertices is NP-complete in a general graph [2]. However, if the input graph is constrained to be planar and k is fixed, then the induced cycle problem can be solved in polynomial time [11, 12, 13]. In particular, an O(n 2) time algorithm is given for the case k = 2 by McDiarmid, Reed, Schrijver and Shepherd [14], where n is the number of vertices of the input graph. Our main results in this paper are to improve their result in the following sense. 1. The number of vertices k is allowed to be non-trivially super constant number, up to k = o((log n log log n) 2 3). More precisely, when k = o((log n log log n) 2 3), then the ICP can be solved in O(n 2+ε) time for any ε > 0. 2. The time complexity is linear if k is fixed. We note that the linear time algorithm (the second result) is independent from the first result. Let us observe that if k is as a part of the input, then the problem is still NP-complete. We need to impose some condition on k.

Lecture Notes in Computer Science, 2008

We present several polynomial-time algorithms for c-planarity testing for clustered graphs with clusters of size at most three. The most general result concerns a special class of Eulerian graphs, namely graphs obtained from a fixed-size 3-connected graph by ...

Rocky Mountain Journal of Mathematics, 2018

We classify all finite groups with planar cyclic graphs. Also, we compute the genus and crosscap number of some families of groups (by knowing that of the cyclic graph of particular proper subgroups in some cases). 1. Introduction. Let G be a group. For each x ∈ G, the cyclizer of x is defined as Cyc G (x) = {y ∈ G | ⟨x, y⟩ is cyclic}. In addition, the cyclizer of G is defined by Cyc(G) = ∩ x∈G Cyc G (x). Cyclizers were introduced by Patrick and Wepsic in [15] and studied in [1, 2, 3, 9, 14, 15]. It is known that Cyc(G) is always cyclic and that Cyc(G) ⊆ Z(G). In particular, Cyc(G) G. The cyclic graph (respectively, weak cyclic graph) of a group G is the simple graph with vertex-set G\Cyc(G) (respectively, G\{1}) such that two distinct vertices x and y are adjacent if and only if ⟨x, y⟩ is cyclic. The cyclic graph and weak cyclic graph of G are denoted by Γ c (G) and Γ w c (G), respectively. From the explanation above, Γ c (G) (respectively, Γ w c (G)) is the null graph if G is cyclic (respectively, trivial). Thus, we will assume that G is non-cyclic (respectively, nontrivial) when working with Γ c (G) (respectively, Γ w c (G)). A graph is planar if it can be drawn in the plane in such a way that its edges intersect only at the end vertices. Recall that a subdivision of an edge {u, v} in a graph Γ is the replacement of the edge {u, v} in Γ with two new edges {u, w} and {w, v} in which w is a new vertex.