| Title: | Induced matching vs edge open packing: trees and product graphs |
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| Authors: | ID Brešar, Boštjan (Author)
ID Dravec, Tanja (Author)
ID Hedžet, Jaka (Author)
ID Samadi, Babak (Author) |
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| Files: |  RAZ_Bresar_Bostjan_2025.pdf (1,31 MB)
MD5: 5063B28258C11C9B62A33E78975BA693
 https://doi.org/10.1016/j.disc.2025.114458
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| Language: | English |
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| Work type: | Scientific work |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: | FNM - Faculty of Natural Sciences and Mathematics
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| 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, ▫$\rho_{e}^{o} (G)$▫, 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 ▫$\rho_{e}^{o}(G)\ge \nu_I(G)$▫ 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. |
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| Keywords: | induced matching, edge open packing, graph product, independent set, trees |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Submitted for review: | 15.09.2024 |
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| Article acceptance date: | 22.02.2025 |
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| Publication date: | 04.03.2025 |
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| Year of publishing: | 2025 |
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| Number of pages: | 19 str. |
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| Numbering: | Letn. 348, št. 7, št. članka 114458 |
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| PID: | 20.500.12556/DKUM-91972  |
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| UDC: | 519.17 |
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| ISSN on article: | 0012-365X |
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| COBISS.SI-ID: | 228279299 |
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| DOI: | 10.1016/j.disc.2025.114458 |
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| Publication date in DKUM: | 25.07.2025 |
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| Views: | 183 |
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| Downloads: | 10 |
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| Metadata: |    |
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| Categories: | Misc.
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