  Given a graph $G$, the maximum size of an induced subgraph of $G$ each
component of which is a star is called the edge open packing number,
$\rho_{e}^{o}(G)$, of $G$. Similarly, the maximum size of an induced subgraph
of $G$ each component of which is the star $K_{1,1}$ is the induced matching
number, $\nu_I(G)$, of $G$. While the inequality $\rho_e^o(G)\geq \nu_{I}(G)$
clearly holds for all graphs $G$, we provide a structural characterization of
those trees that attain the equality. We prove that the induced matching number
of the lexicographic product $G\circ H$ of arbitrary two graphs $G$ and $H$
equals $\alpha(G)\nu_I(H)$. By similar techniques, we prove sharp lower and
upper bounds on the edge open packing number of the lexicographic product of
graphs, which in particular lead to NP-hardness results in triangular graphs
for both invariants studied in this paper. For the direct product $G\times H$
of two graphs we provide lower bounds on $\nu_I(G\times H)$ and
$\rho_{e}^{o}(G\times H)$, both of which are widely sharp. We also present
sharp lower bounds for both invariants in the Cartesian and the strong product
of two graphs. Finally, we consider the edge open packing number in hypercubes
establishing the exact values of $\rho_e^o(Q_n)$ when $n$ is a power of $2$,
and present a closed formula for the induced matching number of the rooted
product of arbitrary two graphs over an arbitrary root vertex.
