On embedding a cycle in a plane graph

Abstract

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.

Clustered Planarity: Embedded Clustered Graphs with Two-Component Clusters

“…It was proved in an equivalent version by Feng et al [5]. A set S of candidate edges satisfying the above conditions is called a saturator 1 . A set S that satisfies the first condition will be called a partial saturator.…”

“…Even in this restricted setting, the complexity of c-planarity testing is unknown. However, polynomial-time algorithms exist for special types of flat clustered graphs, e.g., if the underlying graph is a cycle and the clusters are arranged in a cycle [2], if the underlying graph is a cycle and the clusters are arranged into an embedded plane graph [1], or if the underlying graph is a cycle and the clusters contain at most three vertices [9]. Even for these very restricted settings, the algorithms are quite non-trivial.…”

Clustered Planarity: Small Clusters in Cycles and Eulerian Graphs

“…Another approach is to impose even more restrictions on the structure of the cluster hierarchy. Polynomial-time algorithms exist for flat cluster hierarchy and the following conditions: the underlying graph is a cycle and clusters are arranged in a cycle [3]; the underlying graph is a cycle and clusters are arranged into an embedded plane graph [4].…”

Crossing Minimization in Perturbed Drawings

“…In this paper, we focus on the special case that G is a cycle. A series of recent papers [1,6,8] show that weak embeddings can be recognized in O(n log n) time. Chang et al [6] identified two features of a map ϕ : G → R 2 that are difficult to handle: A spur is a vertex whose incident edges are mapped to the same arc or overlapping arcs, and a fork is a vertex mapped to the relative interior of the image of some nonincident edge (a vertex may be both a fork and a spur).…”