Gap-planar graphs

Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, 2011

Given an n-vertex graph G, a drawing of G in the plane is a mapping of its vertices into points of the plane, and its edges into continuous curves, connecting the images of their endpoints. A crossing in such a drawing is a point where two such curves intersect. In the Minimum Crossing Number problem, the goal is to find a drawing of G with minimum number of crossings. The value of the optimal solution, denoted by OPT, is called the graph's crossing number. This is a very basic problem in topological graph theory, that has received a significant amount of attention, but is still poorly understood algorithmically. The best currently known efficient algorithm produces drawings with O(log 2 n)• (n + OPT) crossings on bounded-degree graphs, while only a constant factor hardness of approximation is known. A closely related problem is Minimum Planarization, in which the goal is to remove a minimum-cardinality subset of edges from G, such that the remaining graph is planar. Our main technical result establishes the following connection between the two problems: if we are given a solution of cost k to the Minimum Planarization problem on graph G, then we can efficiently find a drawing of G with at most poly(d) • k • (k + OPT) crossings, where d is the maximum degree in G. This result implies an O(n • poly(d) • log 3/2 n)approximation for Minimum Crossing Number, as well as improved algorithms for special cases of the problem, such as, for example, k-apex and bounded-genus graphs.

Computational Geometry, 2015

We initiate the study of the following problem: Given a non-planar graph G and a planar subgraph S of G, does there exist a straight-line drawing Γ of G in the plane such that the edges of S are not crossed in Γ by any edge of G? We give positive and negative results for different kinds of connected spanning subgraphs S of G. Moreover, in order to enlarge the subset of instances that admit a solution, we consider the possibility of bending the edges of G not in S; in this setting we discuss different trade-offs between the number of bends and the required drawing area.

2013

We initiate the study of the following problem: Given a non-planar graph G and a planar subgraph S of G, does there exist a straight-line drawing Γ of G in the plane such that the edges of S are not crossed in Γ by any edge of G? We give positive and negative results for different kinds of connected spanning subgraphs S of G. Moreover, in order to enlarge the subset of instances that admit a solution, we consider the possibility of bending the edges of G not in S; in this setting we discuss different trade-offs between the number of bends and the required drawing area.

Journal of Combinatorial Theory, Series B, 1974

In drawings (two edges have at most one point in common, either a node or a crossing) of the complete graph K, in the Euclidean plane there occur at most 2n -2 edges without crossings. This was proved by G. Ringel in [l!. Here the minimal number of edges without crossings in drawings of K,, is determined, and for the existence of values between minimum and maximum is asked. He also remarks in [l], that he knows a special D(K,,) in which no edge without crossings occurs. Here we will show that drawings D(K,J of this kind only exist if n >, 8. Moreover we will determine h(n) for the remaining values of n. 299

JURNAL ILMIAH SAINS

K-CROSSING CRITICAL ALMOST PLANAR GRAPHS ABSTRACT A graph is a pair of a non-empty set of vertices and a set of edges. Graphs can be drawn on the plane with or without crossing of its edges. Crossing number of a graph is the minimal number of crossing among all drawings of the graph on the plane. Graphs with crossing number zero are called planar. A graph is crossing critical if deleting any of its edge decreases its crossing number. A graph is called almost planar if deleting one edge makes the graph planar. This research shows graphs, given an integer k ≥ 1, to build an infinite family of crossing critical almost planar graphs having crossing number k. Keywords: Almost planar graph,crossing critical graph. GRAF K-PERPOTONGAN KRITIS HAMPIR PLANAR ABSTRAK Sebuah graf adalah pasangan himpunan tak kosong simpul dan himpunan sisi. Graf dapat digambar pada bidang dengan atau tanpa perpotongan. Angka perpotongan adalah jumlah perpotongan terkecil di antara semua gambar graf pada bida...

SIAM Journal on Computing, 1988

The pair (G, D) consisting of a planar graph G V, E) with n vertices together with a subset of d special vertices D V is called k-planar if there is an embedding of G in the plane so that at most k faces of G are required to cover all of the vertices in D. Checking 1-planarity can be done in linear-time since it reduces to a problem of checking planarity of a related graph. We present an algorithm which given a graph G and a value k either determines that G is not k-planar or generates an appropriate embedding and associated minimum cover in O(ckn) time, where c is a constant. Hence, the algorithm runs in linear time for any fixed k. The fact that the time required by the algorithm grows exponentially in k is to be expected since we also show that for arbitrary k, the associated decision problem is strongly NP-complete, even when the planar graph has essentially a unique planar embedding, d 0(n), and all facial cycles have bounded length. These results provide a polynomial-time recognition algorithm for special cases of Steiner tree problems in graphs which are solvable in polynomial time. Key words, complexity, planar graphs, Steiner trees AMS(MOS) subject classifications. 05, 68 1. Introduction. Recently, there has been a great deal of interest in solving the Steiner tree problem in graphs. This problem is NP-complete even for planar grid graphs [GJ1]. (See [GJ2] for an excellent introduction to the area of computational complexity.) So recent work has centered on efficiently-solvable special cases and heuristic methods; see [Wi] for a survey of work on this problem. Throughout this paper we deal with undirected graphs of the form G (V, E), where V is a set of n vertices and E is a set of edges connecting pairs of vertices. A graph is called planar if it can be embedded in the plane. A graph G V, E) together with d special vertices D V is called k-planar if there is a 131anar embedding of G so that at most k faces of G are required to cover all of the vertices in D. Clearly, a planar graph is the same as an n-planar graph. The planarity number of G is the minimum k such that G is k-planar. A recent paper by [EMV] presents an algorithm which solves the Steiner problem in an arbitrary graph; their algorithm runs in polynomial time for k-planar graphs, for any fixed k, with D being the vertices required to be in the Steiner tree. It is easy to see that checking 1-planarity of G V, E) with special vertices D V is equivalent to testing the planarity of the associated graph G*= (V*, E*), where V*= Vt.J {r} and E* E [_J {(r, v)" v D}, and so can be done in linear time [HT2]. They leave as an open question the complexity of testing k-planarity for fixed k->-2. In 2, we present an algorithm which checks to see if a given (G, D) pair is k-planar given a fixed embedding of G and if so, determines the planarity number of G in O(ckn) time, when c is a constant. This is used in 3 to generate an appropriate embedding of G and a cover of D by k or fewer faces, if possible, in O(ckn) time. Hence, the algorithm runs in linear time for any fixed k. The fact that the time required grows exponentially in k is to be expected as we show in 4 that for arbitrary k, the associated decision problem is strongly NP-complete, even when the planar graph has essentially a unique planar embedding, d O(n), and all facial cycles have bounded length.

Discrete Applied Mathematics, 2007

The k-planar crossing number of a graph is the minimum number of crossings of its edges over all possible drawings of the graph in k planes. We propose algorithms and methods for k-planar drawings of general graphs together with lower bound techniques. We give exact results for the k-planar crossing number of K 2k+1,q , for k 2. We prove tight bounds for complete graphs. We also study the rectilinear k-planar crossing number.

Journal of Graph Algorithms and Applications, 2013

A straight-line drawing of a planar graph G is a planar drawing of G, where each vertex is mapped to a point on the Euclidean plane and each edge is drawn as a straight line segment. A segment in a straight-line drawing is a maximal set of edges that form a straight line segment. A minimum-segment drawing of G is a straightline drawing of G, where the number of segments is the minimum among all possible straight-line drawings of G. In this paper we prove that it is NP-complete to determine whether a plane graph G has a straight-line drawing with at most k segments, where k ≥ 3. We also prove that the problem of deciding whether a given partial drawing of G can be extended to a straight-line drawing with at most k segments is NP-complete, even when G is an outerplanar graph. Finally, we investigate a worst-case lower bound on the number of segments required by straight-line drawings of arbitrary spanning trees of a given planar graph.

SIAM Journal on Computing, 2013

A graph is near-planar if it can be obtained from a planar graph by adding an edge. We show the surprising fact that it is NP-hard to compute the crossing number of near-planar graphs. A graph is 1-planar if it has a drawing where every edge is crossed by at most one other edge. We show that it is NP-hard to decide whether a given near-planar graph is 1-planar. The main idea in both reductions is to consider the problem of simultaneously drawing two planar graphs inside a disk, with some of its vertices fixed at the boundary of the disk. This leads to the concept of anchored embedding, which is of independent interest. As an interesting consequence we obtain a new, geometric proof of NP-completeness of the crossing number problem, even when restricted to cubic graphs. This resolves a question of Hliněný.

Combinatorica, 1997

We show that if a graph of v vertices can be drawn in the plane so that every edge crosses at most k>0 others, then its number of edges cannot exceed 4.108V"kv. For k<4, we establish a better bound, (kq-3)(v-2), which is tight for k-= 1 and 2. We apply these estimates to improve a result of Ajtai et al. and Leighton, providing a general lower bound for the crossing number of a graph in terms of its number of vertices and edges.