Former Carnegie Mellon math logic doctoral student Peter Glenn returned in early 2024 in hobby mode to studying esoteric math puzzles, in this case the fascinating 2006 proof by Alcon et al that "Clique Graph" is NP-complete. With much labor and practice, he wrote a thirty-page draft paper setting out to explain the 2006 proof with more diagrams and a simpler presentation, his audience "learning alongside" him. As this involved designing "widgets"---small sets of vertices and matching edges that accomplished some sort of logical decision via the "Helly Property"---he came up against graphs he was unsure were correct by manual verification. Moreover he found the graph proposed in the 2006 paper to be wildly untamed for a new researcher like himself. So he wrote a tool set in Python beginning with "checkrs" that verified a given "cover" of a graph was indeed an "RS-cover" (that is, that it covered all edges and consisted of complete sets of vertices from the graph). With this came immediate need for algorithms, as the original runtimes easily led into multiple days needed to compute, say, a seven variable "3SAT" reduction (the genius of the paper being that such a reduction is possible).

Two codebases, Python and C++, are in prototype form on GitHub, each thousands of lines in length. The project has become an occasion to re-learn object-oriented software design whilst also finding highly efficient algorithms (speedups on the order of 99% in some cases, with the code still correct).

Tomorrow's tool is now here today: "Flag Calc", a versatile computer-assistant in the field of basic graph theory, and of extremal asymptotic graph theory. With a notion of "fingerprinting" a graph, a linear ordering is established and quite speedy algorithms are obtained around questions of automorphism counts, of isomorphisms or lack thereof, and of "embeddings" of a subgraph in a graph.

Along the way came a code base around parsing sentences and formulae in standard mathematical logic and algebra. This parameterizable interface immediate verifies things like Mantel's Theorem and many of the proven theorems in any standard graph theory text. The hope is that by "steering" the parameters, the ultra-quick algorithms will give the researcher datum to investigate further, such as new bounds and asymptotic results. But more than this, the heady material does not detract from the usefulness of the tool for all ages, to experiment with unpacking isomorphisms between seemingly-different graphs, and counting out things like "radius" or "diameter", "circumference" and "k-connectedness".