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Via Minimization
Published in Michael Pecht, Placement and Routing of Electronic modules, 2020
Guoqing Li, Michael Pecht, Yeun Tsun Wong
A geometric dual graph Gd = (Nd, Ed) is generated from G(L), where each node in Nd corresponds to a face (cycle) and each edge in Ed corresponds to two adjacent faces in G(L). Let map be a mapping between edge (x,y) ∊ Gd and edge (r,s) ∊ G(L) be cut by (x,y). The node in Gd corresponding to an odd cycle in G(L) is called an odd node. The number of odd nodes is even because the number of odd cycles is even. Thus, removal of an edge in G(L) corresponds to contracting an edge in Gd. When two nodes are merged to contract an edge in Gd, the new node is even only if these two nodes are odd.
Planar Graphs
Published in Kenneth H. Rosen, Graphs, Algorithms, and Optimization, 2005
Given a geographic map drawn in the plane, one can construct a dual graph, by placing a vertex in the interior of each region, and joining vertices by edges if they correspond to adjacent regions. Coloring the regions of the map is then equivalent to coloring the vertices of the dual, so that adjacent vertices are of different colors. Consequently, we shall be concerned with coloring the vertices of a planar graph.
A topological characterisation of looped drainage networks
Published in Structure and Infrastructure Engineering, 2022
Didrik Meijer, Hans Korving, François Clemens-Meyer
Over the past 10–15 years, the dual graph approach has been used to analyse urban road, drainage and water distribution networks. A dual graph or line graph (Harary & Norman, 1960) is a graph in which nodes are replaced by edges and edges by nodes. In a dual graph, each vertex represents an edge of the graph. The dual graph has an edge for each pair of vertices in G that are separated from each other by an edge (Harary & Norman, 1960).