Scopus

Critical Paired Dominating Sets and Irreducible Decompositions of Powers of Edge Ideals

Năm XB 2019 Tạp chí / Hội thảo Acta Mathematica Vietnamica Volume 44 (3) DOI / Link https://doi.org/10.1007/s40306-018-0273-0 ↗

Tác giả

Tóm tắt

Let G be a finite simple graph. A set S of vertices is a critical paired dominating set of G, if every vertex is adjacent to a vertex in S and the removal of any vertex does not change the matching number of G. In this paper, we give a characterization of graphs G which has a critical paired dominating set in terms of the irreducible decomposition of powers of the edge ideal associated to G.

Tài liệu tham khảo

[1] Chen, J., Morey, S., Sung, A.: The stable set of associated primes of the ideal of a graph. Rocky Mountain J. Math. 32(1), 71–89 (2002)

[2] Lou, D., Rao, D.: Characterizing factor critical graphs and an algorithm. Australas. J. Combin. 30, 51–56 (2004)

[3] Edmonds, J.: Paths, trees and flowers. Canad. J. Math. 17, 449–467 (1965)

[4] Gallai, T.: Neuer Beweis eines Tutte’schen Satzes. (German) Magyar Tud. Akad. Mat. Kutató. Int. Kö,zl. 8, 135–139 (1963)

[5] Hà, H. T., Morey, S.: Embedded associated primes of powers of square-free monomial ideals. J. Pure Appl. Algebra 214(4), 301–308 (2010)

[6] Herzog, J., Hibi, T.: The depth of powers of an ideal. J. Algebra 291(2), 534–550 (2005)

[7] Herzog, J., Hibi, T.: Bounding the socles of powers of squarefree monomial ideals. Commutative Algebra and Noncommutative Algebraic Geometry. Vol. II. Math. Sci. Res. Inst. Publ., vol. 68, pp 223–229. Cambridge Univ. Press, New York (2015)

[8] Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of domination in graphs. Marcel Dekker, New York (1998)

[9] Hien, H.T.T., Lam, H.M.: Combinatorial characterizations of the saturation and the associated primes of the fourth power of edge ideals. Acta Math. Vietnam. 40(3), 511–526 (2015)

[10] Haynes, T.W., Slater, P.J.: Paired–domination and the paired–domatic number. Congr. Numer. 109, 65–72 (1995)

[11] Haynes, T.W., Slater, P.J.: Paired–domination in graphs. Networks 32(3), 199–206 (1998)

[12] Lovász, L.: A note on a factor-critical graphs. Studia Sci. Math. Hungar. 7, 279–280 (1972)

[13] Lovász, L., Plummer, M.D.: Matching theory. North-Holland Publishing Co., Amsterdam (1986)

[14] Martinez-Bernal, J., Morey, S., Villarreal, R.: Associated primes of powers of edge ideals. Collect. Math. 63(3), 361–374 (2012)

[15] Miller, E., Sturmfels, B.: Combinatorial commutative algebra graduate texts in mathematics, vol. 227. Springer-Verlag, New York (2005)

[16] Noether, E.: Idealtheorie in Ringbereichen. Math. Ann. 83, 24–66 (1921)

[17] Proffitt, K.E., Haynes, T.W., Slater, P.J.: Paired–domination in grid graphs. Congr. Numer. 150, 161–172 (2001)

[18] Shan, E., Kang, L., Henning, M.A.: A characterization of trees with equal total domination and paired–domination numbers. Australas. J. Combin. 30, 31–39 (2004)

[19] Terai, N., Trung, N.V.: On the associated primes and the depth of the second power of a squarefree monomial ideals. J. Pure Appl. Algebra 218(6), 1117–1129 (2014)

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