The confusing part is the description of how the edges are added and removed to make a "consistent" graph. Graphs don't have a metric associated with the edges but he goes back and forth between the graph representation and referencing a metric structure between the nodes and either adding or removing the shortest path. It makes sense but it's not clear why. I couldn't think of obvious counter-examples to the claim but maybe someone else can. The graphs he draws don't make it obvious that the edges are "directed" (inside vs outside requires choosing a direction/orientation). But after deciding the direction of each edge why does removing the shortest one make sense for computing the correct set of nodes for the union of some given geometric elements?
The confusing part is the description of how the edges are added and removed to make a "consistent" graph. Graphs don't have a metric associated with the edges but he goes back and forth between the graph representation and referencing a metric structure between the nodes and either adding or removing the shortest path. It makes sense but it's not clear why. I couldn't think of obvious counter-examples to the claim but maybe someone else can. The graphs he draws don't make it obvious that the edges are "directed" (inside vs outside requires choosing a direction/orientation). But after deciding the direction of each edge why does removing the shortest one make sense for computing the correct set of nodes for the union of some given geometric elements?