We present a linear programming based algorithm for computing a spanning tree T of a set P of n points in IR, such that its crossing number is O(min(t log n, n1−1/d)), where t the minimum crossing number of any spanning tree of P . This is the first guaranteed approximation algorithm for this problem. We provide a similar approximation algorithm for the more general settings of building a spann...

We introduce a new polynomial invariant of virtual knots and links and use this invariant to compute a lower bound on the virtual crossing number and the minimal surface genus. 1 The arrow polynomial We introduce the arrow polynomial, an invariant of oriented virtual knots and links that is equivalent to the simple extended bracket polnomial [6]. This invariant takes values in the ring Z[A,A, K...

We present a randomized polynomial-time approximation algorithm for the fixed linear crossing number problem (FLCNP). In this problem, the vertices of a graph are placed in a fixed order along a horizontal “node line” in the plane, each edge is drawn as an arc in one of the two half-planes (pages), and the objective is to minimize the number of edge crossings. FLCNP is NP-hard, and no previous ...

CrossingNumber is one of the most challenging algorithmic problems in topological graph theory, with applications to graph drawing and VLSI layout. No polynomial time approximation algorithm is known for this NP-Complete problem. We give in this paper a polynomial time approximation algorithm for the crossing number of toroidal graphs with bounded degree. In course of proving the algorithm we p...

Background Given that school-age students, as active road users, are more vulnerable to injury compared with other pedestrians, a large number of them, following an injury, may either fail to go to school at least for a short time or even suffer from disabilities for the rest of their lives. The aim of this study was to determine safe behavior in road crossing using the theory of planned behavi...

The graph crossing number problem, cr(G) ≤ k, asks for a drawing of a graph G in the plane with at most k edge crossings. Although this problem is in general notoriously difficult, it is fixedparameter tractable for the parameter k [Grohe]. This suggests a closely related question of whether this problem has a polynomial kernel, meaning whether every instance of cr(G) ≤ k can be in polynomial t...

It was proved by [Garey and Johnson, 1983] that computing the crossing number of a graph is an NP -hard problem. Their reduction, however, used parallel edges and vertices of very high degrees. We prove here that it is NP -hard to determine the crossing number of a simple cubic graph. In particular, this implies that the minor-monotone version of crossing number is also NP -hard, which has been...

The problem of drawing a graph in the plane with a minimum number of edge crossings—called the crossing number of a graph—is a well-studied problem which dates back to the first half of the twentieth century, as mentioned in [11], and was formulated in full generality in [3]. It was shown that this problem is NP-Complete [4], and that it remains so even when restricted to cubic graphs [5]. Many...

The graph crossing number problem, cr(G) ≤ k, asks for a drawing of a graph G in the plane with at most k edge crossings. Although this problem is in general notoriously difficult, it is fixedparameter tractable for the parameter k [Grohe]. This suggests a closely related question of whether this problem has a polynomial kernel, meaning whether every instance of cr(G) ≤ k can be in polynomial t...