Domination-Related Parameters in Rooted Product Graphs
Tóm tắt
A set S of vertices of a graph G is a dominating set in G if every vertex outside of S is adjacent to at least one vertex belonging to S. A domination parameter of G is related to those sets of vertices of a graph satisfying some domination property together with other conditions on the vertices of G. Here, we investigate several domination-related parameters in rooted product graphs.
Tài liệu tham khảo
Berge, C.: Graphs and Hypergraphs. North-Holland, Amsterdam (1973)
Brešar, B.: Vizing-like conjecture for the upper domination of Cartesian products of graphs. Electron. J. Comb. 12, N12 (2005)
Brešar, B., Klavžar, S., Rall, D.F.: Dominating direct products of graphs. Discrete Math. 307, 1636–1642 (2007)
Cockayne, E.J., Dreyer, P.A., Hedetniemi, S.M., Hedetniemi, S.T.: Roman domination in graphs. Discrete Math. 278(1–3), 11–22 (2004)
Dorbec, P., Gravier, S., Klavžar, S., Špacapan, S.: Some results on total domination in direct product graphs. Discuss. Math. Graph Theory 26, 103–112 (2006)
Došlić, T., Ghorbani, M., Hosseinzadeh, M.A.: The relationships between Wiener index, stability number and clique number of composite graphs. Bull. Malays. Math. Sci. Soc. 36, 165–172 (2013)
Dunbar, J., Grossman, J., Hedetniemi, S., Hatting, J., McRae, A.: On weakly-connected domination in graphs. Discrete Math. 167–168, 261–269 (1997)
Godsil, C.D., McKay, B.D.: A new graph product and its spectrum. Bull. Australas. Math. Soc. 18(1), 21–28 (1978)
Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. Marcel Dekker, Inc., New York (1998)
Jha, P.K.: Smallest independent dominating sets in Kronecker products of cycles. Discrete Appl. Math. 113, 303–306 (2001)
Klavžar, S., Zmazek, B.: On a Vizing-like conjecture for direct product graphs. Discrete Math. 156, 243–246 (1996)
Lemańska, M., Swaminathan, V., Venkatakrishnan, Y. B., Zuazua, R.: Super dominating sets in graphs. Proc. Natl. Acad. Sci. India Sect. A Phys. Sci. in press (2015)
Lemańska, M.: Weakly convex and convex domination numbers. Opusc. Math. 24(2), 181–188 (2004)
Lemańska, M.: Lower bound on the weakly connected domination number of a tree. Australas. J. Comb. 37, 67–71 (2007)
Rall, D.F.: Total domination in categorical products of graphs. Discuss. Math. Graph Theory 25, 35–44 (2005)
Sampathkumar, E., Walikar, H.: The connected domination number of a graph. J. Math. Phys. Sci. 13(6), 607–613 (1979)
Stewart, I.: Defend the Roman empire!. Sci. Am. 281, 136–138 (1999)
Šumenjak, T.Kraner, Pavlič, P., Tepeh, A.: On the roman domination in the lexicographic product of graphs. Discrete Appl. Math. 160(13–14), 2030–2036 (2012)
Topp, J.: Private communication (2002)
Vizing, V.G.: The Cartesian product of graphs. Vyčisl. Sistemy 9, 30–43 (1963)
Vizing, V.G.: Some unsolved problems in graph theory. Uspehi Math. Nauk 23(144), 117–134 (1968)
Yero, I.G., Rodríguez-Velázquez, J.A..: Roman domination in Cartesian product graphs and strong product graphs. Appl. Anal. Discrete Math. 7, 262–274 (2013)
Yero, I.G., Kuziak, D., Rondón-Aguilar, A.: Coloring,location and domination of corona graphs. Aequ. Math. 86, 1–21 (2013)
Zhao, Y., Shan, E., Liang, Z., Gao, R.: A labeling algorithm for distance domination on block graphs. Bull. Malays. Math. Sci. Soc, in press (2013)
Zwierzchowski, M.: A note on domination parameters of the conjunction of two special graphs. Discuss. Math. Graph Theory 21, 303–310 (2001)