GLOBAL OFFENSIVE k-ALLIANCE IN BIPARTITE GRAPHS
نویسندگان
چکیده
Let k ≥ 0 be an integer. A set S of vertices of a graph G = (V (G), E(G)) is called a global offensive k-alliance if |N(v) ∩ S| ≥ |N(v) − S| + k for every v ∈ V (G) − S, where 0 ≤ k ≤ ∆ and ∆ is the maximum degree of G. The global offensive k-alliance number γ o (G) is the minimum cardinality of a global offensive k-alliance in G. We show that for every bipartite graph G and every integer k ≥ 2, γ o (G) ≤ n(G)+|Lk(G)| 2 , where Lk(G) is the set of vertices of degree at most k− 1. Moreover, extremal trees attaining this upper bound are characterized.
منابع مشابه
Independence and global offensive alliance in graphs
Let G be a simple graph with vertex set V (G). A non-empty set S ⊆ V (G) is a global strong offensive alliance if for every vertex v in V (G)−S, a strict majority of its closed neighborhood is in S. The global strong offensive alliance number γô(G) is the minimum cardinality of a global strong offensive alliance of G. We show that if G is a connected bipartite graph of order at least three, the...
متن کاملBounds on the global offensive k-alliance number in graphs
Let G = (V (G), E(G)) be a graph, and let k ≥ 1 be an integer. A set S ⊆ V (G) is called a global offensive k-alliance if |N(v) ∩ S| ≥ |N(v)−S|+k for every v ∈ V (G)−S, where N(v) is the neighborhood of v. The global offensive k-alliance number γ o (G) is the minimum cardinality of a global offensive k-alliance in G. We present different bounds on γ o (G) in terms of order, maximum degree, inde...
متن کاملConnected global offensive k-alliances in graphs
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 graphG, ifG[S] is connected and |N(v)∩S| ≥ |N(v)−S|+k for every vertex v ∈ V (G...
متن کاملGlobal offensive alliance numbers in graphs with emphasis on trees
For a graph G = (V,E), a non-empty set S ⊆ V is a global offensive alliance if for every v ∈ V − S, at least half of the vertices from the closed neighborhood of v are in S. A set S ⊆ V is a global strong offensive alliance if for each vertex v ∈ V −S, a strict majority of the vertices of the closed neighborhood of v are in S. The cardinality of a minimum global (strong) offensive alliance of a...
متن کاملComputing global offensive alliances in Cartesian product graphs
A global offensive alliance in a graph G is a set S of vertices with the property that every vertex not belonging to S has at least one more neighbor in S than it has outside of S. The global offensive alliance number of G, γo(G), is the minimum cardinality of a global offensive alliance in G. A set S of vertices of a graph G is a dominating set for G if every vertex not belonging to S has at l...
متن کامل