Research about crossings is typically about minimization. In this paper, we consider maximizing the number of crossings over all possible ways to draw a given graph in the plane. Alpert et al. [1] conjectured that any graph has a convex straight-line drawing, e.g., a drawing with vertices in convex position, that maximizes the number of edge crossings. We disprove this conjecture by constructin...

This note outlines the realizable extension problem for weighted graphs and provides results of a detailed analysis of this problem for the weighted graph (K3,3, l). This analysis is then utilized to provide a result relating to the connectedness of the moduli space of planar realizations of (K3,3, l). The note culminates with two examples which show that in general, realizability and connected...

We present a method to find closed feature lines which indicate sharp edges in a point cloud in the context of reverse engineering. We start with a first order segmentation which results in different point clusters. A weighted graph structure is built where vertices correspond to the resulting point clusters and edges connect neighboring clusters. By choosing the weights carefully, the minimum ...

• The all-pairs shortest paths problem on weighted digraphs (APSP). • Detecting if a weighted graph has a triangle of negative total edge weight. • Listing up to n2.99 negative triangles in an edge-weighted graph. • Finding a minimum weight cycle in a graph of non-negative edge weights. • The replacement paths problem on weighted digraphs. • Finding the second shortest simple path between two n...

Consider a bipartite graph; letÕs suppose we draw the origin nodes and the destination nodes arranged in two columns, and the edges as straight line segments. A non crossing matching is a subset of edges such that no two of them intersect. Several algorithms for the problem of finding the non crossing matching of maximum cardinality are proposed. Moreover an extension to weighted graphs is cons...

The problem of optimal space allocation among interconnecting wires of VLSI chips, in order to minimize their switching power consumption is solved. Necessary and sufficient conditions for the existence of optimal space allocation are derived, stating that every wire must be in equilibrium of its line-to-line weighted capacitance density on its two opposite sides. Two proofs are presented, one ...

In this paper, we propose a new type of graph, denoted as ”embedded-graph”, and its theory, which employs a distributed representation to describe the relations on the graph edges. Embedded-graphs can express linguistic and complicated relations, which cannot be expressed by the existing edge-graphs or weighted-graphs. We introduce the mathematical definition of embedded-graph, translation, edg...

an extended star is a tree which has only one vertex with degree larger than two. the -center problem in a graph  asks to find a subset  of the vertices of  of cardinality  such that the maximum weighted distances from  to all vertices is minimized. in this paper we consider the -center problem on the unweighted extended stars, and present some properties to find solution.

Python implementation of selected weighted graph algorithms is presented. The minimal graph interface is defined together with several classes implementing this interface. Graph nodes can be any hashable Python objects. Directed edges are instances of the Edge class. Graphs are instances of the Graph class. It is based on the adjacency-list representation, but with fast lookup of nodes and neig...

Given a weighted planar graph, we construct a simple separating cycle C whose interior and exterior both weigh at most 2/3 of the total weight. The cycle C uses k jumps across the faces and ordinary edges with cost O(1=k) of the total edge cost. This theorem is the main component of a PTAS for the weighted TSP problem 4].