Abstract
We describe the class of graphs whose every subgraph has the next property: The maximal number of disjoint 4-paths is equal to the minimal cardinality of sets of vertices such that every 4-path in the subgraph contains at least one of these vertices.We completely describe the set of minimal forbidden subgraphs for this class. Moreover, we present an alternative description of the class based on the operations of edge subdivision applied to bipartite multigraphs and the addition of the so-called pendant subgraphs, isomorphic to triangles and stars.
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References
V. E. Alekseev and D. B. Mokeev, “König Graphs with Respect to 3-Paths,” Diskretn. Anal. Issled. Oper. 19 (4), 3–14 (2012).
F. Kardoš, J. Katrenič, and I. Schiermeyer, “On Computing the Minimum 3-Path Vertex Cover and Dissociation Number of Graphs,” Theor. Comput. Sci. 412 (50), 7009–7017 (2011).
Y. Li and J. Tu, “A 2-Approximation Algorithm for the Vertex Cover P4Problem in Cubic Graphs,” Int. J. Comput.Math. 91 (10), 2103–2108 (2014).
D. S. Malyshev, “The Impact of the Growth Rate of the Packing Number of Graphs on the Computational Complexity of the Independent Set Problem,” Diskretn. Mat. 25 (2), 63–67 (2013) [Discrete Math. Appl. 23 (3–4), 245–249 (2013)].
P. Hell, “Graph Packings,” Electron. Notes DiscreteMath. 5, 170–173 (2000).
R. Yuster, “Combinatorial and Computational Aspects of Graph Packing and Graph Decomposition,” Comput. Sci. Rev. 1 (1), 12–26 (2007).
B. Brešar, F. Kardoš, J. Katrenič, and G. Semanišin, “Minimum k-Path Vertex Cover,” Discrete Appl.Math. 159 (12), 1189–1195 (2011).
J. Tu and W. Zhou, “A Primal-Dual Approximation Algorithm for the Vertex Cover P3Problem,” Theor. Comput. Sci. 412 (50), 7044–7048 (2011).
J. Edmonds, “Paths, Trees, and Flowers,” Can. J. Math. 17 (3–4), 449–467 (1965).
D. G. Kirkpatrick and P. Hell, “On the Completeness of a Generalized Matching Problem,” in Proceedings of 10th Annual ACM Symposium on Theory of Computing (San Diego, USA, May 1–3, 1978) (ACM, New York, 1978), pp. 240–245.
S. Masuyama and T. Ibaraki, “Chain Packing in Graphs,” Algorithmica 6 (1), 826–839 (1991).
N. S. Devi, A. C. Mane, and S. Mishra, “Computational Complexity of Minimum P4Vertex Cover Problem for Regular and K1,4-Free Graphs,” Discrete Appl.Math. 184 (12), 114–121 (2015).
V. E. Alekseev and D. B. Mokeev, “König Graphs for 3-Paths and 3-Cycles,” Discrete Appl.Math. 204, 1–5 (2016).
R.W. Deming, “IndependenceNumbers ofGraphs–an Extension of the König–Egervary Theorem,” Discrete Math. 27, 23–33 (1979).
A. Kosowski, M. Małafiejski, and P. Żyličski, “Combinatorial and Computational Aspects of Graph Packing and Graph Decomposition,” Graphs Combin. 24 (5), 461–468 (2008).
F. Sterboul, “A Characterization ofGraphs inWhich the TransversalNumber Equals theMatching Number,” J.Combin. Theory Ser. B 27 (2), 228–229 (1979).
D. B. Mokeev, “On König Graphs with Respect to P4,” Diskretn. Anal. Issled. Oper. 24 (3), 61–79 (2017) [J. Appl. Indust. Math. 11 (3), 421–430 (2017)].
G. Ding, Z. Xu, and W. Zang, “Packing Cycles in Graphs, II,” J. Combin. Theory Ser. B 87 (2), 244–253 (2003).
R. Diestel, Graph Theory (Springer, Heidelberg, 2005).
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Russian Text © D.S. Malyshev, D.B. Mokeev, 2019, published in Diskretnyi Analiz i Issledovanie Operatsii, 2019, Vol. 26, No. 1, pp. 74–88.
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Malyshev, D.S., Mokeev, D.B. König Graphs with Respect to the 4-Path and Its Spanning Supergraphs. J. Appl. Ind. Math. 13, 85–92 (2019). https://doi.org/10.1134/S1990478919010101
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DOI: https://doi.org/10.1134/S1990478919010101