Midterm ish

My summer of code is broken up into several projects. There were a lot of small ones, a couple medium ones, and one large one. Right now, I'm in the midst of working on the large project. Basically, we want to feed Sage a collection of subsets of an edge set E, and have Sage tell us if there is a graph that has cycles which correspond to the subsets of E, and if so, to give a corresponding. This boils down to asking if a matroid is graphic, and asking for a graph that realizes the matroid.

For instance, if we give have E = {1, 2, 3, 4}, and our collection of sets is any three element subset of E, then we can't get an appropriate graph. To see this, we start constructing a graph. Our first cycle is {1, 2, 3}, There is only one graph on three elements that has this cycle, namely a triangle. To add the edge 4, we need to have a cycle {1, 2, 4}. But this means that we have to add 4 in parallel to the edge 3. This is a problem, because then {1, 3, 4}, in particular, is not a cycle of our graph.

This example illustrates a key idea of the algorithm. The set {1, 2} is a maximal set that is not contained in a cylce, so we skipped over those elements, and started with 3. We then added 3 and any needed elements of {1, 2} to our partial graph. And we kept adding elements till we either had a problem, or till we added all of the elements.

In our case, we didn't get so complicated of a graph that we had a choice about which graph to use for our partial graph. In general, this is not the case. It would be troublesome to check if we could add the new element to every graphs that realizes the already added elements, so we use a decomposition made possible by Whitney's 2 isomorphism theorem to check all of the graphs options at once. This of course makes the code more complicated. The algorithm that we are following comes from a paper by Ronald Bixby and Donald Wagner.

The tricky part, so far, has been trying to get information in and out of graphs. graph theorists care a lot about the vertices of a graph and much less about the edges of the graph. That is, they store their edges as a list of the two vertices that they are incident with, and a possible label. matroid theorists, however, care a lot more about the edges of a graph. This is true in general, and is true in particular for this project.