Review article
Parameterized analysis and crossing minimization problems☆

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Abstract

In this survey/introductory article, we first present the basics of the field of Parameterized Complexity, made accessible to readers without background on the subject. Afterwards, we survey some central questions in Graph Drawing that concern the analysis of crossing minimization problems from the viewpoint of parameterized analysis, as well as put forward some of the remaining challenges. This article originated from an invited talk given at the 29th International Symposium on Graph Drawing and Network Visualization.

Section snippets

Background on parameterized complexity

Towards the introduction of some basic concepts in Parameterized Complexity, let us start with an example. In (the classic parameterized version of) the Vertex Cover problem, we are given an undirected graph G and a non-negative integer k∈N0. The objective is to determine whether G has a vertex cover—that is, a subset of vertices U⊆V(G) such that for every edge {u,v}∈E(G), {u,v}∩U≠0̸—of size at most k. While the Vertex Cover problem is NP-hard, we may still desire to solve it. A natural way to

The classic crossing number problem

In the last few years, there has been a substantial increase in the number of works at the intersection of Parameterized Complexity and Graph Drawing—specifically, problems in graph drawing studied from the perspective of parameterized analysis. For a few illustrative examples, let us mention that this includes studies of crossing minimization (on which we elaborate below), recognition of planar graph families such as upward planarity testing [40], [41] and grid graph recognition [42], as well

Restricted embeddings

Over the years, several restricted variants of the Crossing Number problem where drawings must have some particular form have been introduced and studied. Here, we consider three such prominent restrictions that are of interest from the viewpoint of Parameterized Complexity.

Fixed point sets/embeddings: Computation of subgraphs with few crossings and related variants

Settings where we are given a set of points P in the plane that represent vertices, and edges are to be drawn as straight lines between them, are intensively studied since the early 80s. A large body of works has been devoted to the establishment of combinatorial bounds on the number of crossing-free graphs on P, where particular attention is given to crossing-free triangulations, perfect matchings and Hamiltonian paths and cycles. Originally, the study of these bounds was initiated by Newborn

Testing k-planarity and related variants

A k-planar graph is a graph that admits a plane drawing where every edge is crossed at most k times. Notice that in the last paragraph of Section 3.2, we have considered a notion similar to that of a k-planar graph, where the drawing was restricted to be a book embedding. Testing even 1-planarity (where given a graph G, we need to decide whether it is 1-planar) is NP-hard [135], [136], even when restricted to almost-planar graphs [55], or to graphs of bounded (by a constant) bandwidth [43]

Crossing number as a parameter

So far, we have discussed only the computation of drawings of graphs—or the computation of subgraphs of already drawn graphs—that have few crossings. In most cases, the crossing number (or variant of it) has been part of the parameterization. However, to date, various other computational problems have been already studied on graphs with few crossings. For example, Grigoriev and Bodlaender [135] considered graphs that can be embedded on a surface of bounded genus such that each edge has a

Variants of crossing number as a measure

Over the years, a wide-variety of notions based on the crossing number have been introduced and extensively studied. Here, we survey a few of them, with emphasis on the perspective of Parameterized Complexity.

Odd Crossing Number. The odd crossing number is defined similarly to the crossing number, or, even more similarly, to the pairwise crossing number, but it only counts pairs of distinct edges that intersect in an odd number of points. The computation of the odd crossing number of a given

Conclusion

The main purpose of this article has been to provide a survey of the current state-of-the-art of the study of crossing minimization problems from the viewpoint of Parameterized Complexity, with emphasis on: (1) attraction of researchers outside the Parameterized Complexity community to consider the perspective of parameterized analysis in the study of graph drawing problems, by the inclusion of a gentle introduction to the area of Parameterized Complexity; and, in particular, (2) suggestion of

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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    ☆

    Supported by the Israel Science Foundation (ISF) under the Individual Research Grant program (grant no. 1176/18), the European Research Council (ERC) Starting Grant (PARAPATH), and the United States – Israel Binational Science Foundation (BSF) Start Up Grant (grant no. 2018302).

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