Global defensive alliances in the lexicographic product of paths and cycles
نویسندگان
چکیده
منابع مشابه
On global (strong) defensive alliances in some product graphs
A defensive alliance in a graph is a set $S$ of vertices with the property that every vertex in $S$ has at most one moreneighbor outside of $S$ than it has inside of $S$. A defensive alliance $S$ is called global if it forms a dominating set. The global defensive alliance number of a graph $G$ is the minimum cardinality of a global defensive alliance in $G$. In this article we study the global ...
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A defensive alliance in a graph G = (V,E) is a set of vertices S ⊆ V satisfying the condition that for every vertex v ∈ S, the number of neighbors v has in S plus one (counting v) is at least as large as the number of neighbors it has in V − S. Because of such an alliance, the vertices in S, agreeing to mutually support each other, have the strength of numbers to be able to defend themselves fr...
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We consider complexity issues and upper bounds for defensive alliances and strong global offensive alliances in graphs. We prove that it is NP-complete to decide for a given 6-regular graph G and a given integer a, whether G contains a defensive alliance of order at most a. Furthermore, we prove that determining the strong global offensive alliance number γô(G) of a graph G is APX-hard for cubi...
متن کاملGlobal defensive k-alliances in graphs
Let Γ = (V,E) be a simple graph. For a nonempty set X ⊆ V , and a vertex v ∈ V , δX(v) denotes the number of neighbors v has in X. A nonempty set S ⊆ V is a defensive k-alliance in Γ = (V,E) if δS(v) ≥ δS̄(v)+k, ∀v ∈ S. A defensive k-alliance S is called global if it forms a dominating set. The global defensive k-alliance number of Γ, denoted by γ k(Γ), is the minimum cardinality of a defensive ...
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A defensive alliance in a graph G = (V,E) is a set of vertices S ⊆ V where for each v ∈ S, at least half of the vertices in the closed neighborhood of v are in S. A defensive alliance S is called global if every vertex in V (G) \S is adjacent to at least one member of the defensive alliance S. In this paper, we derive an upper bound to the size of the minimum global defensive alliances in star ...
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ژورنال
عنوان ژورنال: Discrete Applied Mathematics
سال: 2020
ISSN: 0166-218X
DOI: 10.1016/j.dam.2019.12.024