Let $D$ be a $(v,k,\lambda)$-design. By the **domination number of $D$** I mean the domination number $\gamma(L(D))$ of the bipartite incidence graph of $D$.



> 
> Is $\gamma(L(D))$ determined only by $v,k$, and $\lambda$,
>  irrespective of the actual structure of $D$?
> 
> 
> 


I can prove this for finite projective planes and have empirically verified this property for $(8,4,3)$ and $(10,4,2)$-designs (of which there are four and three non-isomorphic ones, respectively).


P.S.
Has the domination number of such graphs been studied at all? I could only find one paper by Laskar *et al.* which considered the line graphs of the incidence graphs.



Is the domination number of a combinatorial design determined by the design parameters?
Well, interestingly enough, all the 80 Steiner triple systems on 15 points have minimum dominating sets of size 10 - there is more tradeoff between the points/blocks than I first recognised.


But if we go a bit bigger then we can find some variation. Here is a $2$-$(25,4,1)$ design with 50 blocks.



```
0 1 2 3; 0 4 5 6; 0 7 8 9; 0 10 11 12; 0 13 14 15; 0 16 17 18; 0 19 20 21; 0 22 23 24; 1 4 7 10; 1 5 8 13; 1 6 14 16; 1 9 11 19; 1 12 15 22; 1 17 20 23; 1 18 21 24; 2 4 8 23; 2 5 7 21; 2 6 12 24; 2 9 10 17; 2 11 14 18; 2 13 20 22; 2 15 16 19; 3 4 13 18; 3 5 11 22; 3 6 7 19; 3 8 12 17; 3 9 15 24; 3 10 16 20; 3 14 21 23; 4 9 16 22; 4 11 20 24; 4 12 14 19; 4 15 17 21; 5 9 14 20; 5 10 15 18; 5 12 16 23; 5 17 19 24; 6 8 15 20; 6 9 18 23; 6 10 21 22; 6 11 13 17; 7 11 15 23; 7 12 18 20; 7 13 16 24; 7 14 17 22; 8 10 14 24; 8 11 16 21; 8 18 19 22; 9 12 13 21; 10 13 19 23;

```

and here is another



```
0 1 2 3; 0 4 5 6; 0 7 8 9; 0 10 11 12; 0 13 14 15; 0 16 17 18; 0 19 20 21; 0 22 23 24; 1 4 7 10; 1 5 8 13; 1 6 16 19; 1 9 14 22; 1 11 20 23; 1 12 17 21; 1 15 18 24; 2 4 11 18; 2 5 7 24; 2 6 8 21; 2 9 16 20; 2 10 15 19; 2 12 13 22; 2 14 17 23; 3 4 12 23; 3 5 15 20; 3 6 7 17; 3 8 10 22; 3 9 11 19; 3 13 18 21; 3 14 16 24; 4 8 14 20; 4 9 15 17; 4 13 19 24; 4 16 21 22; 5 9 12 18; 5 10 16 23; 5 11 14 21; 5 17 19 22; 6 9 13 23; 6 10 14 18; 6 11 15 22; 6 12 20 24; 7 11 13 16; 7 12 14 19; 7 15 21 23; 7 18 20 22; 8 11 17 24; 8 12 15 16; 8 18 19 23; 9 10 21 24; 10 13 17 20;

```

My computer tells me that the first one has a dominating set of size 13, namely the $9$ points $$\{1,2,7,12,14,17,18,20,22\}$$ and the four blocks $$\{0,4,5,6 \mid 3, 9 ,15, 24 \mid 8 ,11, 16 ,21 \mid 10, 13, 19, 23\}.$$ You can check this by hand and probably also convince yourself that no fewer than $13$ will do. 


However the computer also tells me that the second design has no dominating set of size $13$, but how you would convince yourself of this by hand is another matter.


P.S While I was at it, I tried the $960$ $2$-$(10,3,2)$ designs (which have 30 blocks so are a bit smaller) and determined that $42$ have domination number $6$, and the remainder $7$.


