Geodesic
Global alliances and independence in trees
Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 1, pp. 19-27.

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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.
Keywords: defensive alliance, offensive alliance, global alliance, domination, trees, independence number 
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Chellali, Mustapha; Haynes, Teresa. Global alliances and independence in trees. Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 1, pp. 19-27. http://geodesic.mathdoc.fr/item/DMGT_2007_27_1_a1/

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