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1

Starting with 5,7,$f$ and $g$ is correct. In order to complete the maximum independent you can proceed as follows. First, delete the vertices that are reachable from an unmatched vertex via an alternating path (in your example 5,6,7,$e$,$f$,$g$). This leaves a balanced bipartite graph (that is, it has the same number of vertices on each side), and you can ...

3

Here's an argument that you need time quadratic in the number of polygons. More precisely, you should not be able to find containing pairs among $n$ $k$-sided polygons in time $O(n^{2-\epsilon})$, for any $k=O(n^{\delta})$ and any $\epsilon,\delta>0$ It's a reduction from the orthogonal vectors problem, the problem of finding two disjoint binary vectors ...

2

First you can read the CGAL documentation that can help you to understand combinatorial and generalized maps, since it provide several examples. You can also read the book "Combinatorial Maps: Efficient Data Structures for Computer Graphics and Image Processing". Now some answers: 1) I don't think. In a combinatorial map, in 2D, a triangle is described by ...

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