Secure Sets and Defensive Alliances in Graphs: A Faster Algorithm and Improved Bounds
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 ...
متن کاملGlobal Defensive Alliances in Graphs
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...
متن کاملDefensive k-alliances in graphs
Let Γ = (V,E) be a simple graph of order n and degree sequence δ1 ≥ δ2 ≥ · · · ≥ δn. 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. The defensive k-alliance number of Γ, denoted by ak(Γ), is defined as the minimum cardinality of a defensive k-alliance in Γ...
متن کاملExact Defensive Alliances in Graphs
A nonempty set S ⊂ V is a defensive k-alliance in G = (V,E), k ∈ [−Δ,Δ] ∪ Z, if for every v ∈ S, dS(v) ≥ dS̄(v) + k. A defensive k-alliance S is called exact, if S is defensive k-alliance but is no defensive (k+1)-alliance in G. In this paper we study the mathematical properties of exact defensive k-alliances in graphs. In particular, we obtain several bounds for defensive k-alliance of a graph....
متن کاملOn defensive alliances and line graphs
Let Γ be a simple graph of size m and degree sequence δ1 ≥ δ2 ≥ · · · ≥ δn. Let L(Γ) denotes the line graph of Γ. The aim of this paper is to study mathematical properties of the alliance number, a(L(Γ), and the global alliance number, γa(L(Γ)), of the line graph of a simple graph. We show that ⌈
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ژورنال
عنوان ژورنال: IEICE Transactions on Information and Systems
سال: 2015
ISSN: 0916-8532,1745-1361
DOI: 10.1587/transinf.2014fcp0007