We introduce the Orchard crossing number, which is defined in a similar way to the well-known rectilinear crossing number. We compute the Orchard crossing number for some simple families of graphs. We also prove some properties of this crossing number. Moreover, we define a variant of this crossing number which is tightly connected to the rectilinear crossing number, and compute it for some sim...

Rectilinear Binary Space Partitioning (BSP) trees are often used for solving various types of range searching problems including ray shooting. We propose a novel method for construction of rectilinear BSP trees for a preferred set of ray shooting queries. Particularly, we study ray sets formed by fixing either the direction or the origin of rays. We analyse and discuss the properties of constru...

We show that a metric space embeds in the rectilinear plane (i.e., is L 1-embeddable in R 2) if and only if every subspace with five or six points does. A simple construction shows that for higher dimensions k of the host rectilinear space the number c(k) of points that need to be tested grows at least quadratically with k, thus disproving a conjecture of Seth and Jerome Malitz.

In the rectilinear Steiner arborescence problem the task is to build a shortest rectilinear Steiner tree connecting a given root and a set of terminals which are placed in the plane such that all root-terminal-paths are shortest paths. This problem is known to be NP-hard. In this paper we consider a more restricted version of this problem. In our case we have a depth restrictions d(t) ∈ N for e...

The planar rectilinear Steiner tree problem has been extensively studied. The common formulation ignores circuit fabrication issues such as multiple routing layers, preferred routing directions, and vias between layers. In this paper, the authors extend a previously presented planar rectilinear Steiner tree heuristic to consider layer assignment, preferred routing direction restrictions, and vi...

The dilation of a geometric graph is the maximum, over all pairs of points in the graph, of the ratio of the Euclidean length of the shortest path between them in the graph and their Euclidean distance. We consider a generalized version of this notion, where the nodes of the graph are not points but axis-parallel rectangles in the plane. The arcs in the graph are horizontal or vertical segments...

We extend known results regarding the maximum rectilinear crossing number of the cycle graph (Cn) and the complete graph (Kn) to the class of general d-regular graphs Rn,d. We present the generalized star drawings of the d-regular graphs Sn,d of order n where n + d ≡ 1 (mod 2) and prove that they maximize the maximum rectilinear crossing numbers. A star-like drawing of Sn,d for n ≡ d ≡ 0 (mod 2...

The spanning ratio and maximum detour of a graph G embedded in a metric space measure how well G approximates the minimum complete graph containing G and metric space, respectively. In this paper we show that computing the spanning ratio of a rectilinear path P in L1 space has a lower bound of Ω(n log n) in the algebraic computation tree model and describe a deterministic O(n log n) time algori...

We consider the following problem known as simultaneous geometric graph embedding (SGE). Given a set of planar graphs on a shared vertex set, decide whether the vertices can be placed in the plane in such a way that for each graph the straight-line drawing is planar. We partially settle an open problem of Erten and Kobourov [5] by showing that even for two graphs the problem is NP-hard. We also...