TITLE: What is the smallest graph that contains all non-isomorphic 4-node and 5-node connected graphs as induced subgraphs?
QUESTION [6 upvotes]: I'm looking for an undirected graph $G$ which contains as induced subgraphs:

the six $4$-node connected graphs and,
the twenty one $5$-node connected graphs.

Ideally, I would also like it to be as small as possible (least number of nodes).  If we take the union of these graphs, it has $6 \times 4+21 \times 5=129$ nodes.  If we take the graphs and identify them at a point, it has $6 \times 4+21 \times 5-(6+21)+1=103$ nodes.
This would be helpful for testing our program for network motif detection program (NetMODE), which searches for connected induced subgraphs in networks.
If it helps, I've attached a picture of the graphs below:

REPLY [6 votes]: The size of a graph that contains all $k$-vertex subgraphs as induced subgraphs is exponential in $k$.
There are more than $2^{\binom{k}2}/k!$ graphs on $n$ vertices. So if $X$ is a graph on $n$ vertices
that contains all $k$ vertex subgraphs as induced subgraphs then
$$
 \binom{n}{k} \ge 2^{\binom{k}2}/k!
$$
and as
$$
        \binom{n}{k} \le \frac{n^k}{k!}
$$
we have
$$
 n^k \ge 2^{\binom{k}2}.
$$
So
$$
 k\, \log_2(n) \ge k(k-1)/2
$$
and therefore $n \ge 2^{(k-1)/2}$.
Let $p$ be a prime congruent to 1 mod 4. The vertices of the Paley graph
are the integers mod $p$; two integers are adjacent if their difference is
a square mod $p$. The Paley graph on $p$ vertices contains all graphs on $k$
vertices as induced subgraphs, where $k\approx c\,\log_2(p)$.