Abstract
The dominance complex D(G) of a simple graph \(G = (V,E)\) is the simplicial complex consisting of the subsets of V whose complements are dominating. We show that the connectivity of D(G) plus 2 is a lower bound for the vertex cover number \(\tau (G)\) of G.
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References
Adamaszek, M.: Splittings of independence complexes and the powers of cycles. J. Comb. Theory Ser. A 119, 1031–1047 (2012)
Babson, E., Kozlov, D.N.: Proof of the Lovász conjecture. Ann. Math. 165, 965–1007 (2007)
Barmak, J.A.: Star clusters in independence complexes of graphs. Adv. Math. 241, 33–57 (2013)
Deshpande, P., Singh, A.: Higher independence complexes of graphs and their homotopy types. to appear in Journal of the Ramanujan Mathematical Society (2021)
Dochtermann, A.: Exposed circuits, linear quotients, and chordal clutters. J. Comb. Ser. Theory A 177, 105327 (2021)
Ehrenborg, R., Hetyei, G.: The topology of independence complex. Eur. J. Comb. 27, 906–923 (2006)
Engström, A.: Complexes of directed trees and independence complexes. Discrete Math. 309, 3299–3309 (2009)
Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2001)
Jonsson, J.: On the topology of independence complexes of triangle-free graphs. Preprint
Kawamura, K.: Independence complexes of chordal graphs. Discrete Math. 310, 2204–2211 (2010)
Kozlov, D.N.: Complexes of directed trees. J. Comb. Theory Ser. A 88, 112–122 (1999)
Marietti, M., Testa, D.: A uniform approach to complexes arising from forests. Electron. J. Comb. 15 (2008)
Marietti, M., Testa, D.: Cores of simplicial complexes. Discrete Comput. Geom. 40, 444–468 (2008)
Matoušek, J.: Using the Borsuk–Ulam Theorem. Lectures on Topological Methods in Combinatorics and Geometry, 2nd edn. Springer, Universitext (2007)
Meshulam, R.: Domination numbers and homology. J. Comb. Theory Ser. A 102, 321–330 (2003)
Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra. Graduate Texts in Mathematics, vol. 227. Springer, New York (2005)
Milnor, J.W., Stasheff, J.D.: Characteristic Classes. Princeton University Press, Princeton (1974)
Nagel, U., Reiner, V.: Betti numbers of monomial ideals and shifted skew shapes. Electron. J. Comb. 16 (2009)
Taylan, D.: Matching trees for simplicial complexes and homotopy type of devoid complexes of graphs. Order 33, 459–476 (2016)
Tsukuda, S.: Independence complexes and incidence graphs. Contrib. Discret. Math. 12(1), 28–46 (2015)
Woodroofe, R.: Chordal and sequentially Cohen–Macaulay clutters. Electron. J. Comb. 18 (2011)
Acknowledgements
The author is partially supported by JSPS KAKENHI 19K14536. The author thanks the referee for useful comments. The author states that there is no conflict of interests.
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Matsushita, T. Dominance complexes and vertex cover numbers of graphs. J Appl. and Comput. Topology 7, 363–368 (2023). https://doi.org/10.1007/s41468-022-00109-2
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DOI: https://doi.org/10.1007/s41468-022-00109-2