In this paper we investigate the computational complexity of the following problem. Given a planar graph in which some vertices have already been placed in the plane (i.e., a partial embedding), place the remaining vertices to form a planar straight-line drawing of the graph. We show that the above extensibility problem, mentioned in the 2003 paper “Selected Open Problems in Graph Drawing” [1],...

We present a new method for solving symbolically zero–dimensional polynomial equation systems in the affine and toric case. The main feature of our method is the use of problem adapted data structures : arithmetic networks and straight–line programs. For sequential time complexity measured by network size we obtain the following result : it is possible to solve any affine or toric zero–dimensio...

The classical Fáry’s theorem from the 1930s states that every planar graph can be drawn as a straight-line drawing. In this paper, we extend Fáry’s theorem to non-planar graphs. More specifically, we study the problem of drawing 1-planar graphs with straight-line edges. A 1-planar graph is a sparse non-planar graph with at most one crossing per edge. We give a characterisation of those 1planar ...

The results of patients operated on by Nepal Cleft Lip and Palate Association (NECLAPA) surgeons for congenital macrostomia were prospectively studied between January 2000 and December 2002. There were four males and three females with a median age of 10 years. Three had an associated branchial arch syndrome. In all patients an overlapping repair of orbicularis oris was done. Six patients had a...

We investigate periodic straight-line orbits (SLO) in Hamiltonian force fields using both direct and inverse methods. A general theorem is proven for natural Hamiltonians quadratic in the momenta in arbitrary dimension and specialized to two and three dimension. Next we specialize to homogeneous potentials and their superpositions, including the familiar HénonHeiles problem. It is shown that SL...

This paper studies the problem of computing 3D crossing-free straight-line grid drawings of graphs such that the overall volume is small. We show that every 2-tree (and therefore every series-parallel graph) can be drawn on an integer 3D grid consisting of 15 parallel lines and having linear volume. We extend the study to the problem of drawing general k-trees on a set of parallel grid lines. L...

We show that certain explicit operators must vastly increase the straight line complexity of certain polyno-mials.

graphs are denoted by capital roman letters and plane graphs by capital greek letters. If the graph G is isomorphic to the plane graph F, then F is a representation of G. If the edges of F are polygonal arcs (respectively straight line segments), then F is a polygonal arc representation (respectively straight line representation). A closed Jordan curve partitions the Euclidean plane into a boun...

It is shown that every connected planar straight line graph with n ≥ 3 vertices has an embedding preserving augmentation to a 2-edge connected planar straight line graph with at most b(2n − 2)/3c new edges. It is also shown that every planar straight line tree with n ≥ 3 vertices has an embedding preserving augmentation to a 2-edge connected planar topological graph with at most bn/2c new edges...

Execution speed of programs on modern computer architectures is sensitive, by a factor of two or more, to the order in which instructions are presented to the processor. To realize potential execution efficiency, it is now customary for an optimizing compiler to employ a heuristic algorithm for instruction scheduling. These algorithms are painstakingly hand-crafted, which is expenseive and time...