A linear arrangement / of an undirected graph G = (V,E) with jVj = n nodes is a bijective function /:V ? {0, . . . , n 1}. The cost function is costðG;/Þ 1⁄4 P uv2Ejð/ðuÞ /ðvÞÞj and opt(G) = min"/cost(G,/). The problem of finding opt(G) is called minimum linear arrangement (MINLA). The Minimum Linear Arrangement is an NP-hard problem in general. But there are some classes of graphs optimally so...

Introduction. A pseudoline is formed from a line by stretching the plane without tearing: it is the image of a line under a homeomorphism of the plane [13]. In arrangements of pseudolines, pairs of pseudolines intersect at most once and cross at their intersections. Pseudoline arrangements can be used to model sorting networks [1], tilings of convex polygons by rhombi [4], and graphs that have ...

We generalize the Hadwiger theorem on line transversals to collections of compact convex sets in the plane to the case where the sets are connected and the transversals form an arrangement of pseudolines. The proof uses the embeddability of pseudoline arrangements in topological affine planes. In 1940 Santaló showed [8], by an example, that Vincensini’s proof [9] of an extension of Helly’s theo...

Let G be a graph embedded in the plane and let A be an arrangement of pseudolines intersecting the drawing of G. An aligned drawing of G and A is a planar polyline drawing Γ of G with an arrangement A of lines so that Γ and A are homeomorphic to G and A. We show that if A is stretchable and every edge e either entirely lies on a pseudoline or intersects at most one pseudoline, then G and A have...

A linear arrangement is a labeling or a numbering or a linear ordering of the vertices of a graph. In this paper we solve the minimum linear arrangement problem for bijective connection graphs (for short BC graphs) which include hypercubes, Möbius cubes, crossed cubes, twisted cubes, locally twisted cube, spined cube, Z-cubes, etc. as the subfamilies.

Let G be an undirected graph with vertex set V(G) and edge set E(G) (see [2] for undefined graph theory terminology). A linear arrangement n of G is a one-to-one mapping from V(G) to the set of positive integers. The weight of an edge e = {u, u} in a linear arrangement 7c is defined to be the absolute value of z(u) n(v). The cost of the linear arrangement 7~ of G, denoted by J=(C), is defined t...

Refer to Figure 1. A monotone matching is a set of n segments, each the portion of a unique pseudoline, and each spanning a unique slab, such that the left endpoint of each segment is above the right endpoint of the segment in the previous slab. In addition, the point on the first vertical line is below the left endpoint of the first segment, and the point on the last vertical line is above the...