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The bipartite crossing number problem is studied and a connection between this problem and the linear arrangement problem is established. A lower bound and an upper bound for the optimal number of crossings are derived, where the main terms are the optimal arrangement values. Two polynomial time approximation algorithms for the bipartite crossing number are(More)
The antibandwidth problem consists of placing the vertices of a graph on a line in consecutive integer points in such a way that the minimum difference of adjacent vertices is maximized. The problem was originally introduced in [15] in connection with multiprocessor scheduling problems and can be also understood as a dual problem to the well known bandwidth(More)
In the circular (other alternate concepts are outerplanar, convex and one-page) drawing one places vertices of a n−vertex m−edge connected graph G along a circle, and the edges are drawn as straight lines. The smallest possible number of crossings in such a drawing of the graph G is called circular (outerplanar,convex, or one-page) crossing number of the(More)
Let G be a connected bipartite graph.We give a short proof, using a variation of Menger's Theorem, for a new lower bound which relates the bipartite crossing number of G, denoted by bcr(G), to the edge connectivity properties of G. The general lower bound implies a weaker version of a very recent result, establishing a bisection based lower bound on bcr(G)(More)
Accuracy of fit of denture bases is critical to adequate retention. This study compared the dimensional change of a newer continuous-injection technique with a standard trial-pack technique as determined by measuring the posterior palatal border opening. The influence of palate shape and immersion were also assessed. Stone casts were made from master moulds(More)
An outerplanar (also called circular, convex, one-page) drawing of an n-vertex graph G is a drawing in which the vertices are placed on a circle and each edge is drawn using one straight-line segment. We derive exact results for the minimal number of crossings in any outer-planar drawings of the following classes of graphs: 3-row meshes, Halin graphs and(More)