Abstract. For a graph G = (V, E), a set S ⊆ V is a dominating set if every vertex in V − S has at least a neighbor in S. A dominating set S is a global offensive alliance if for every vertex v in V − S, at least half of the vertices in its closed neighborhood are in S. The domination number γ(G) is the minimum cardinality of a dominating set of G and the global offensive alliance number γo(G) is the minimum cardinality of a global offensive alliance of G. We first show that every tree of order at least three with leaves and s support vertices satisfies γo(T ) ≥ (n − + s + 1)/3 and we characterize extremal trees attaining this lower bound. Then we give a constructive characterization of trees with equal domination and global offensive alliance numbers.
For a graph $G=(V,E)$, a set $D\subseteq V$ is a dominating set if every vertex in $V-D$ is either in $D$ or has a neighbor in $D$. A dominating set $D$ is a global offensive alliance (resp. a global defensive alliance) if for each vertex $v$ in $V-D$ (resp. $v$ in $D$) at least half the vertices from the closed neighborhood of $v$ are in $D$. A global powerful alliance is both global defensive and global offensive.
The global powerful alliance number $\gamma_{pa}(G)$ is the minimum cardinality of a global powerful alliance of $G$. We show that if $T$ is a tree of order $n$ with $l$ leaves and $s$ support vertices, then $\gamma_{pa}(T)\geq\frac{3n-2l-s+2}{5}$. Moreover, we provide a constructive characterization of all extremal trees attaining this bound.
For a graph G = (V, E), a set S ⊆ V is a dominating set if every vertex in V − S has at least a neighbor in S. A dominating set S is a global offensive alliance if for each vertex v in V − S at least half the vertices from the closed neighborhood of v are in S. The domination number γ(G) is the minimum cardinality of a dominating set of G, and the global offensive alliance number γo(G) is the minimum cardinality of a global offensive alliance of G. We show that if G is a connected unicycle graph of order n with l(G) leaves and s(G) support vertices then γo(G) ≥ n−l(G)+s(G)
3. Moreover, we characterize all extremal unicycle graphs attaining this bound.
Let G = (V, E) be a simple graph. A non-empty set S ⊆ V is called a global offensive alliance if S is a dominating set and for every vertex v in V − S, at least half of the vertices from the closed neighborhood of v are in S. The global offensive alliance number is the minimum cardinality of a global offensive alliance in G. In this paper, we give a constructive characterization of trees having a unique minimum global offensive alliance.
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