Connected global offensive k-alliances in graphs

• Autorzy: Lutz Volkmann
• Języki publikacji: EN

Abstrakty

EN

We consider finite graphs G with vertex set V(G). For a subset S ⊆ V(G), we define by G[S] the subgraph induced by S. By n(G) = |V(G) | and δ(G) we denote the order and the minimum degree of G, respectively. Let k be a positive integer. A subset S ⊆ V(G) is a connected global offensive k-alliance of the connected graph G, if G[S] is connected and |N(v) ∩ S | ≥ |N(v) -S | + k for every vertex v ∈ V(G) -S, where N(v) is the neighborhood of v. The connected global offensive k-alliance number $γₒ^{k,c}(G)$ is the minimum cardinality of a connected global offensive k-alliance in G.
In this paper we characterize connected graphs G with $γₒ^{k,c}(G) = n(G)$. In the case that δ(G) ≥ k ≥ 2, we also characterize the family of connected graphs G with $γₒ^{k,c}(G) = n(G) - 1$. Furthermore, we present different tight bounds of $γₒ^{k,c}(G)$.

Słowa kluczowe

EN

alliances in graphs; connected global offensive k-alliance; global offensive k-alliance; domination

Kategorie tematyczne

• 05C69: Dominating sets, independent sets, cliques

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2011
• Tom: 31
• Numer: 4
• Strony: 699-707

Daty

wydano

2011

otrzymano

2010-06-11

poprawiono

2010-11-05

zaakceptowano

2010-11-05

Twórcy

autor

Lutz Volkmann

• Lehrstuhl II für Mathematik, RWTH Aachen University, Templergraben 55, D-52056 Aachen, Germany

Bibliografia

• [1] S. Bermudo, J.A. Rodriguez-Velázquez, J.M. Sigarreta and I.G. Yero, On global offensive k-alliances in graphs, Appl. Math. Lett. 23 (2010) 1454-1458, doi: 10.1016/j.aml.2010.08.008.
• [2] M. Chellali, Trees with equal global offensive k-alliance and k-domination numbers, Opuscula Math. 30 (2010) 249-254.
• [3] M. Chellali, T.W. Haynes, B. Randerath and L. Volkmann, Bounds on the global offensive k-alliance number in graphs, Discuss. Math. Graph Theory 29 (2009) 597-613, doi: 10.7151/dmgt.1467.
• [4] H. Fernau, J.A. Rodriguez and J.M. Sigarreta, Offensive r-alliance in graphs, Discrete Appl. Math. 157 (2009) 177-182, doi: 10.1016/j.dam.2008.06.001.
• [5] J.F. Fink and M.S. Jacobson, n-domination in graphs, in: Graph Theory with Applications to Algorithms and Computer Science (John Wiley and Sons, New York, 1985) 283-300.
• [6] J.F. Fink and M.S. Jacobson, On n-domination, n-dependence and forbidden subgraphs, in: Graph Theory with Applications to Algorithms and Computer Science (John Wiley and Sons, New York, 1985) 301-311.
• [7] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998).
• [8] T.W. Haynes, S.T. Hedetniemi, and P.J. Slater, Domination in Graphs: Advanced Topics ( Marcel Dekker, New York, 1998).
• [9] P. Kristiansen, S.M. Hedetniemi and S.T. Hedetniemi, Alliances in graphs, J. Combin. Math. Combin. Comput. 48 (2004) 157-177.
• [10] O. Ore, Theory of graphs (Amer. Math. Soc. Colloq. Publ. 38 Amer. Math. Soc., Providence, R1, 1962).
• [11] K.H. Shafique and R.D. Dutton, Maximum alliance-free and minimum alliance-cover sets, Congr. Numer. 162 (2003) 139-146.
• [12] K.H. Shafique and R.D. Dutton, A tight bound on the cardinalities of maximum alliance-free and minimum alliance-cover sets, J. Combin. Math. Combin. Comput. 56 (2006) 139-145.

Identyfikatory

DOI

10.7151/dmgt.1574