Abstract
In a graph G, a set D of vertices is said to be a monopoly if any vertex \({v\in V(G) \setminus D}\) has at least deg(v)/2 neighbors in D. A strict monopoly is defined similarly when we replace deg(v)/2 by deg(v)/2 + 1 for any vertex v whose degree is even number. By the monopoly size (resp. strict monopoly size) of G we mean the smallest cardinality of a monopoly (resp. strict monopoly) in G. We first discuss the basic bounds for the monopoly and strict monopoly size of graphs. In the second section we show relationships between matchings and monopolies and present some upper bounds for the monopoly and strict monopoly size of graphs in terms of the matching number of graphs. The third section is devoted to presenting some lower bounds for the monopoly size of graphs in terms of the even-girth and odd-girth of graphs.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bermond J.-C., Bond J., Peleg D., Perennes S.: The power of small coalitions in graphs. Discrete Appl. Math. 127, 399–414 (2003)
Bondy J.A., Murty U.S.R.: Graph Theory. Springer, Berlin (2008)
Dreyer P.A., Roberts F.S: Irreversible k-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Appl. Math 157, 1615–1627 (2009)
Favaron, O.: Global alliances and independent domination in some classes of graphs. Electron. J. Combin. 15 (2008) #R123
Favaron O., Fricke G., Goddard W., Hedetniemi S.M., Hedetniemi S.T., Kristiansen P., Laskar R.C., Skaggs R.D: Offensive alliances in graphs. Discuss. Math. Graph Theory 24(2), 263–275 (2004)
Fernau H., Rodriguez J.A., Sigarreta J.M: Offensive r-alliances in graphs. Discrete Appl. Math. 157, 177–182 (2009)
Flocchini P., Kralovic R., Roncato A., Ruzicka P., Santoro N: On time versus size for monotone dynamic monopolies in regular topologies. J. Discrete Algorithms 1, 129–150 (2003)
Kristiansen P., Hedetniemi S.M., Hedetniemi S.T: Alliances in graphs. J. Combin. Math. Combin. Comput. 48, 157–177 (2004)
Lovász, L., Plummer, M.D.: Matching Theory. North Holland, Amsterdam (1986)
Mishra A., Rao S.B: Minimum monopoly in regular and tree graphs. Discrete Math. 306, 1586–1594 (2006)
Peleg D.: Local majorities, coalitions and monopolies in graphs. Theoret. Comput. Sci. 282, 231–257 (2002)
Rodriguez J.A., Sigarreta J.M: Offensive alliances in cubic graphs. Int. Math. Forum 1(36), 1773–1782 (2006)
Sigarreta J.M., Rodriguez J.A: On the global offensive alliance number of a graph. Discrete Appl. Math. 157, 219–226 (2009)
Zaker M.: Bounds for chromatic number in terms of even-girth and booksize. Discrete Math. 311, 197–204 (2011)
Zaker M.: On dynamic monopolies of graphs with general thresholds. Discrete Math. 312, 1136–1143 (2012)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Khoshkhah, K., Nemati, M., Soltani, H. et al. A Study of Monopolies in Graphs. Graphs and Combinatorics 29, 1417–1427 (2013). https://doi.org/10.1007/s00373-012-1214-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-012-1214-7