The idea of the multi-level technique is to reduce the magnitude of a partition on this reduced graph, and finally project this partition on the original graph.
2.
Then adding the edges ( A, D ) and ( B, C ) to the reduced graph yields another spanning tree, and by the triangle inequality it's shorter than the one we started with, which is a contradiction.
3.
In practice, rather than maintain an explicit graph representation for each reduced graph, we will remove vertices from the graph as we go, adding them to a stack, then color them as we pop them back off the stack at the end.
4.
For " st "-planar graphs that are not transitively reduced, an equivalent transitively reduced graph may be obtained by area of the drawings it produces, the number of bends, and the aspect ratio of the drawing, but less well in total edge length.
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