Conjecture 9.6. For all graphs $G$ and $H$, $$\gamma(G \square H) \ge \min\{i(G)\gamma(H), i(H)\gamma(G)\}$$

where $\square$ is the cartesian product of graphs and $i(G)$ is the independent domination number.

For cartesian squares it is $$\gamma(G \square G) \ge \gamma(G) i(G)$$

According to sage and my verification square of the graph on 7 vertices $[0 \ldots 6]$ with edges $$[(0, 4), (0, 5), (0, 6), (1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6)]$$ appears a counterexample to Conjecture 9.6 (note that vertex 3 is disconnected).

Computation:

```
sage: G=Graph(':Fo@I@I@J') #from sparse6
sage: Gs=G.cartesian_product(G)
sage: (Gs.dominating_set(value_only=True),G.dominating_set(value_only=True),G.dominating_set(value_only=True,independent=True) )
(11, 3, 4)
```

Verification:

Since the order is only $7$, $\gamma(G)=3$ and $i(G)=4$ were verified by enumerating all subsets of the vertices. For $\gamma(G \square G)=11$ the dominating set returned by sage was verified and it is an upper bound for the correct value.

I well might have misunderstood the conjecture.

Is the above graph a counterexample of Conjecture 9.6?

Adding disconnected vertices to $G$ gives more counterexamples.