Testing Planarity of Partially Embedded Graphs

Abstract

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.Further, by equipping the components of the decomposition with suitable data structures and by carefully splitting the problem into simpler subproblems, we improve our algorithm to reach linear-time complexity.Finally, we consider several generalizations of the problem, e.g. minimizing the number of edges of the partial embedding that need to be rerouted to extend it, and argue that they are NP-hard. Also, we show how our algorithm can be applied to solve related Graph Drawing problems.

Strip Planarity Testing for Embedded Planar Graphs

“…Several papers in the graph drawing literature (see, e.g., [2,3,29]), however, assume a definition for "embedding of a non-connected planar graph G" in which the containment relationships between distinct connected components of G are prescribed in advance. More formally, let G 1 , .…”

“…Such variants mainly focus on testing, for a given planar graph G, the existence of a planar drawing of G satisfying certain constraints. For example the partial embedding planarity problem [3,29] asks whether a planar drawing G of a given planar graph G exists in which the drawing of a subgraph H of G in G coincides with a given drawing H of H . Clustered planarity testing [14,19,30], upward planarity testing [7,23,27], level planarity testing [31], embedding constrained planarity testing [24], radial level planarity testing [6], and clustered level planarity testing [5,20] are further examples of problems falling in this category.…”

Free Edge Lengths in Plane Graphs

“…Patrignani [16] also showed that it is NP-hard to decide whether a straight-line embedding of a subgraph G (i.e., a partial embedding) can be extended to an embedding of a host H , G ⊂ H . For curvilinear embeddings, this problem is known as planarity testing for partially embedded graphs (Pep), which is decidable in polynomial time [2]. Recently, Jelínek et al [13] gave a combinatorial characterization for Pep via a list of forbidden substructures.…”

A kuratowski-type theorem for planarity of partially embedded graphs

“…Despite of this being a very natural generalization of planarity, this approach has been considered only recently [1]. It should be mentioned that all previous planarity testing algorithms have been of little use for Pep, as they all allow flipping of already drawn parts of the graph, and thus are not suitable for preserving an embedding of a given subgraph.…”

“…In this paper we complement the algorithm in [1] by a study of the combinatorial aspects of this question. In particular, we provide a complete characterization of planar Pegs via a small set of forbidden substructures, similarly to the celebrated Kuratowski theorem [10] that characterizes planarity via the forbidden minors K5 and K3,3.…”