Defensive k-alliances in graphs
نویسندگان
چکیده
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. We study the mathematical properties of ak(Γ). We show that ⌈ δn+k+2 2 ⌉ ≤ ak(Γ) ≤ n− ⌊ δn−k 2 ⌋
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 کاملStrong Alliances in Graphs
For any simple connected graph $G=(V,E)$, a defensive alliance is a subset $S$ of $V$ satisfying the condition that every vertex $vin S$ has at most one more neighbour in $V-S$ than it has in $S$. The minimum cardinality of any defensive alliance in $G$ is called the alliance number of $G$, denoted $a(G)$. In this paper, we introduce a new type of alliance number called $k$-strong alliance numb...
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 کاملUpper k-Alliances in Graphs
A nonempty set S ⊆ V is a defensive k-alliance in Γ = (V,E), k ∈ [−Δ,Δ] ∩ Z, if for every v ∈ S, δS(v) ≥ δS̄(v) + k. An defensive kalliance S is called critical if no proper subset of S is an defensive k-alliances in Γ = (V,E). The upper defensive k-alliance number of Γ, denoted by Ak(Γ), is defined as the maximum cardinality of a critical defensive k-alliance in Γ. In this paper we study the ma...
متن کاملBoundary defensive k-alliances in graphs
We define a boundary defensive k-alliance in a graph as a set S of vertices with the property that every vertex in S has exactly k more neighbors in S than it has outside of S. In this paper we study mathematical properties of boundary defensive k-alliances. In particular, we obtain several bounds on the cardinality of every boundary defensive k-alliance. Moreover, we consider the case in which...
متن کاملDefensive alliances in regular graphs and circulant graphs
In this paper we study defensive alliances in some regular graphs. We determine which subgraphs could a critical defensive alliance of a graph G induce, if G is 6-regular and the cardinality of the alliance is at most 8. In particular, we study the case of circulant graphs, i.e. Cayley graphs on a cyclic group. The critical defensive alliances of a circulant graph of degree at most 6 are comple...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Appl. Math. Lett.
دوره 22 شماره
صفحات -
تاریخ انتشار 2009