TITLE: Induced subgraphs of small strongly regular graphs
QUESTION [7 upvotes]: Consider a strongly regular graph $G$ with parameters $(76,30,8,14).$ Hoffman's bound tells us that $\overline{G}$ has an independent set of size at most $4$ and its not hard to see there are indeed $4$ independent vertices in $\overline{G}.$
So if $x \in V(G)$ then the graph $H$ induced by $N(v)$ is a $8$-regular, $K_4$-free graph on $30$ vertices and known bounds tell us that $\alpha(H) \ge 5.$
Hence $G$ must must contain $K_{1,5}$ as an induced subgraph. I do not see any way to extend this subgraph without introducing cases.
What I am wondering is 

Question 1. Can someone construct larger graphs that must be present as induced subgraphs of $G$

and

Question 2. Can someone find large induced subgraphs for some of the missing SRG's on less than 100 vertices?

The motivating factor for this problem is that getting an induced subgraph of order 19 not having $2$ as an eigenvalue is enough to reconstruct $G.$

REPLY [2 votes]: On strategies of reconstructing bigger subgraphs: suppose
for a vertex $v$ we have we know the set $M$ of (possible) subgraphs $G(v,w)$ induced on $N(v,w):=N(v)\cap N(w)$, for a vertex $w$ at distance 2 from $v$, and the set $\Lambda$ of (possible) subgraphs $G(v,u)$ induced on $N(v,u)$, for $u\in N(v)$.
Take $H\in M$ and $F\in\Lambda$ so that they match on $N(u,v)$ and on $N(u,w)$, for $w\in N(u)-N(v)$ (i.e. we identify $H$ with  $G(v,w)$ and $F$ with $G(v,u)$). Now, let us pick $w\neq w_1\in N(u)-N(v)$, and go though all the possible $H_1\in M$ so that in addition $H_1$ can be identified with $G(v,w_1)$; for each success we can further pick $w_2\in N(u)-N(v)-\{w,w'\}$ and
go through all the possible $H_2\in M$ so that $H_2$ can be identified with $G(v,w_2)$, etc. 
One possible variation here is to switch the roles of $u$ and $v$ and reconstruct subgraphs in $N(u)$ in the same vein.
If such a search tree does not get too big, one can think of what to do next.