The associahedron is a polytope whose graph is the graph of flips on triangulations of a convex polygon. Pseudotriangulations and multitriangulations generalize triangulations in two different ways, which have been unified by Pilaud and Pocchiola in their study of pseudoline arrangements with contacts supported by a given network. In this paper, we construct the “brick polytope” of a network, o...

Tropical oriented matroids were defined by Ardila and Develin in 2007. They are a tropical analogue of classical oriented matroids in the sense that they encode the properties of the types of points in an arrangement of tropical hyperplanes – in much the same way as the covectors of (classical) oriented matroids describe the types in arrangements of linear hyperplanes. Not every oriented matroi...

We show that the optimum shortcut, with respect to routing cost, in a directed path with weighted vertices can be found in linear time. To this end, we construct in linear time the lower envelope of an arrangement of pseudo-lines whose order at infinity is given. The algorithm can also be applied to star-shaped directed networks.

Topological drawings are natural representations of graphs in the plane, where vertices are represented by points, and edges by curves connecting the points. Topological drawings of complete graphs and of complete bipartite graphs have been studied extensively in the context of crossing number problems. We consider a natural class of simple topological drawings of complete bipartite graphs, in ...

For typical #P-hard problems on graphs, we have recently proposed an approach to solve those problems of moderate size rigorously by means of the binary decision diagram, BDD [12, 13]. This paper extends this approach to counting problems on linear matroids, graphic arrangements and partial orders, most of which are already known to be #P-hard, with using geometric properties. E cient algorithm...

Lichen nitidus (LN) is a rare chronic inflammatory disease of unknown etiology. It predominantly affects children and young adults. Its clinical manifestation typically corresponds to erythematous or flesh-colored, asymptomatic or little pruritic mini papules, often located in the trunk, genitalia, and extremities. The linear arrangement of this dermatosis seems rare. To our knowledge, only 2 c...

We prove that the topological complexity of (a motion planning algorithm on) the complement of generic complex essential hyperplane arrangement of n hyperplanes in an r-dimensional linear space is min{n + 1, 2r}.