Induced universal graphs for families of small graphs

Abstract

We present exact and heuristic algorithms that find, for a given family of graphs, a graph that contains each member of the family as an induced subgraph. For $0\le k\le 6$, we give the minimum number of vertices $f(k)$ in a graph containing all $k$-vertex graphs as induced subgraphs, and show that $16\le f(7)\le 18$. For $0\le k\le 5$, we also give the counts of such graphs, as generated by brute-force computer search. We give additional results for small graphs containing all trees on $k$ vertices.

22 pages, 13 figures, 4 tables. This version had additional results and more detailed explanations than the previous version