Lower Bounds for the Thickness and the Total Number of Edge Crossings of Euclidean Minimum Weight Laman Graphs and (2,2)-Tight Graphs
We explore the maximum total number of edge crossings and the maximum geometric thickness of the Euclidean minimum-weight (k, ℓ)-tight graph on a planar point set P. In this paper, we show that (10/7-ε)|P| and (11/6-ε)|P| are lower bounds for the maximum total number of edge crossings for any ε>0...
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| Veröffentlicht in: | IEICE Transactions on Information and Systems 2024/06/01, Vol.E107.D(6), pp.732-740 |
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| container_title | IEICE Transactions on Information and Systems |
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| creator | KAWAKAMI, Yuki TAKAHASHI, Shun SETO, Kazuhisa HORIYAMA, Takashi KOBAYASHI, Yuki HIGASHIKAWA, Yuya KATOH, Naoki |
| description | We explore the maximum total number of edge crossings and the maximum geometric thickness of the Euclidean minimum-weight (k, ℓ)-tight graph on a planar point set P. In this paper, we show that (10/7-ε)|P| and (11/6-ε)|P| are lower bounds for the maximum total number of edge crossings for any ε>0 in cases (k, ℓ)=(2, 3) and (2, 2), respectively. We also show that the lower bound for the maximum geometric thickness is 3 for both cases. In the proofs, we apply the method of arranging isomorphic units regularly. While the method is developed for the proof in case (k, ℓ)=(2, 3), it also works for different ℓ. |
| doi_str_mv | 10.1587/transinf.2023EDP7214 |
| format | Article |
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| source | J-STAGE Free; Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals |
| subjects | (k, ℓ)-tight graph geometric thickness Graphs k-planarity Laman graph Lower bounds Minimum weight sparse graph Thickness |
| title | Lower Bounds for the Thickness and the Total Number of Edge Crossings of Euclidean Minimum Weight Laman Graphs and (2,2)-Tight Graphs |
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