Patrick and I have a long list of -properties and -bounds to code. *Independence Number Theory *be viewed as whatever properties and invariants we’ve already coded. So we can generate graphs which are “difficult” with respect to the coded theory. In our current minimal configuration, a 5-cycle with a chord popped out as difficult. This graph has a vertex *v* such that the graph induced on N[v] is complete. It can be argued that such a vertex is in some maximum independent set. This graph can be reduced. So the graph tells us what -property to code next.

Addendum: this process has continued. With the limited theory coded so far, a graph that was “almost KE” popped out, so that property was added, then a graph that was claw-free, so that property was added. Then the following graph popped out:

Notice that if you remove edge 6-0 you get a KE graph. We call the property where removal of *some* edge yields a KE graph, “almost-edge-KE”. It can be argued that the independence number of an almost-edge-KE graph G can be efficiently computed. Either G has a non-empty KE part, or every non-empty independent set has more neighbors, and . It can then be argued that, if *n* is odd, so n=2k+1, then , and if *n* is even, so n=2k, then