Induced matching vs edge open packing: trees and product graphs
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, $ρ_{e}^{o}(G)$, 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, $ν_I(G)$, of $G$. While the inequality $ρ_e^o(G)\geq ν_{I}(G)$ 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 $α(G)ν_I(H)$. 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 $ν_I(G\times H)$ and $ρ_{e}^{o}(G\times H)$, 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 $ρ_e^o(Q_n)$ 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.