# Global alliances and independence in trees

Mustapha Chellali; Teresa W. Haynes

Discussiones Mathematicae Graph Theory (2007)

- Volume: 27, Issue: 1, page 19-27
- ISSN: 2083-5892

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topMustapha Chellali, and Teresa W. Haynes. "Global alliances and independence in trees." Discussiones Mathematicae Graph Theory 27.1 (2007): 19-27. <http://eudml.org/doc/270383>.

@article{MustaphaChellali2007,

abstract = {A global defensive (respectively, offensive) alliance in a graph G = (V,E) is a set of vertices S ⊆ V with the properties that every vertex in V-S has at least one neighbor in S, and for each vertex v in S (respectively, in V-S) at least half the vertices from the closed neighborhood of v are in S. These alliances are called strong if a strict majority of vertices from the closed neighborhood of v must be in S. For each kind of alliance, the associated parameter is the minimum cardinality of such an alliance. We determine relationships among these four parameters and the vertex independence number for trees.},

author = {Mustapha Chellali, Teresa W. Haynes},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {defensive alliance; offensive alliance; global alliance; domination; trees; independence number},

language = {eng},

number = {1},

pages = {19-27},

title = {Global alliances and independence in trees},

url = {http://eudml.org/doc/270383},

volume = {27},

year = {2007},

}

TY - JOUR

AU - Mustapha Chellali

AU - Teresa W. Haynes

TI - Global alliances and independence in trees

JO - Discussiones Mathematicae Graph Theory

PY - 2007

VL - 27

IS - 1

SP - 19

EP - 27

AB - A global defensive (respectively, offensive) alliance in a graph G = (V,E) is a set of vertices S ⊆ V with the properties that every vertex in V-S has at least one neighbor in S, and for each vertex v in S (respectively, in V-S) at least half the vertices from the closed neighborhood of v are in S. These alliances are called strong if a strict majority of vertices from the closed neighborhood of v must be in S. For each kind of alliance, the associated parameter is the minimum cardinality of such an alliance. We determine relationships among these four parameters and the vertex independence number for trees.

LA - eng

KW - defensive alliance; offensive alliance; global alliance; domination; trees; independence number

UR - http://eudml.org/doc/270383

ER -

## References

top- [1] M. Blidia, M. Chellali and O. Favaron, Independence and 2-domination in trees, Austral. J. Combin. 33 (2005) 317-327.
- [2] G. Chartrand and L. Lesniak, Graphs & Digraphs: Third Edition (Chapman & Hall, London, 1996).
- [3] E.J. Cockayne, O. Favaron, C. Payan and A.G. Thomason, Contributions to the theory of domination, independence, and irredundance in graphs, Discrete Math. 33 (1981) 249-258, doi: 10.1016/0012-365X(81)90268-5. Zbl0471.05051
- [4] T.W. Haynes, S.T. Hedetniemi, and M.A. Henning, Global defensive alliances in graphs, The Electronic J. Combin. 10 (2003) R47. Zbl1031.05096
- [5] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998). Zbl0890.05002
- [6] T.W. Haynes, S.T. Hedetniemi and P.J. Slater (eds), Domination in Graphs: Advanced Topics (Marcel Dekker, New York, 1998). Zbl0883.00011
- [7] S.M. Hedetniemi, S.T. Hedetniemi and P. Kristiansen, Alliances in graphs, J. Combin. Math. Combin. Comput. 48 (2004) 157-177. Zbl1051.05068

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