Induced matching vs edge open packing: trees and product graphs

TL;DR

This work systematically analyzes the induced matching number $\nu_I(G)$ and the edge open packing number $\rho_e^o(G)$ across trees and the four standard graph products. It provides a complete tree-structure characterization for equality $\nu_I(T)=\rho_e^o(T)$, exact formulas for $\nu_I$ under lexicographic products, and sharp bounds for $\rho_e^o$ under the same product, including NP-hardness results for triangular graphs. The paper extends these techniques to the direct, Cartesian, strong, and rooted products, deriving sharp lower bounds, exact hypercube values when $n$ is a power of two, and a closed formula for $\nu_I(G\circ_v H)$ that yields corona-product results. Collectively, these results illuminate how induced matchings and edge open packings behave under product operations and supply precise combinatorial estimates and complexity insights for product graphs.

Abstract

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.

Figures (3)

• Figure 1: An illustration of the situation discussed in Claim \ref{['cl:zadnja']} of the proof of Theorem \ref{['thm:trees2']}.
• Figure 2: The graph $P_{3}\times P_{19}$. The bold edges form an optimal EOP set.
• Figure 3: The graph $G$. The bold edges form a $\rho_{e}^{o}(H)$-set of cardinality $3r+2$.

Theorems & Definitions (42)

• Proposition 2.1
• Proposition 2.2
• Lemma 2.5
• proof
• Theorem 2.6
• proof
• Lemma 2.7
• proof
• Theorem 2.8
• proof