Global defensive alliances in trees - Bouzefrane, Mohamed; Chellali, Mustapha; Haynes, Teresa W. (Utilitas Mathematica, 2010)
A global defensive 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 at least half the vertices from the closed neighborhood of v are in S. The alliance is called strong if a strict majority of vertices from the closed neighborhood of v are in S. The global defensive alliance number γa(G) (respectively, global strong defensive alliance number γâ(G)) is the minimum cardinality of a global defensive alliance (respectively, global strong defensive alliance) of G. We show that if T is a tree with order n ≥ 2, l leaves and s support vertices, then γâ (T) ≥ (3n - l - s + 4) /6 and γa (T) ≥ (3n - l - s + 4)/8. Moreover, all extremal trees attaining each bound are characterized.