Defensive K-alliance Research Articles

On global defensive k-alliances in zero-divisor graphs of finite commutative rings

The global defensive [Formula: see text]-alliance is a very well-studied notion in graph theory, it provides a method of classification of graphs based on relations between members of a particular set of vertices. In this paper, we explore this notion in zero-divisor graph of commutative rings. The established results generalize and improve recent work by Muthana and Mamouni who treated a particular case for [Formula: see text] known by the global defensive alliance. Various examples are also provided which illustrate and delimit the scope of the established results.

Global defensive k-alliances in directed graphs: combinatorial and computational issues

In this paper we define the global defensive k-alliance (number) in a digraph D, and give several bounds on this parameter with characterizations of all digraphs attaining the bounds. In particular, for the case k = −1, we give a lower (an upper) bound on this parameter for directed trees (rooted trees). Moreover, the characterization of all directed trees (rooted trees) for which the equality holds is given. Finally, we show that the problem of finding the global defensive k-alliance number of a digraph is NP-hard for any suitable non-negative value of k, and in contrast with it, we also show that finding a minimum global defensive (−1)-alliance for any rooted tree is polynomial-time solvable.

Distribution of global defensive k-alliances over some graph products

If $$G=(V_G, E_G)$$ is a graph, then $$S\subseteq V_G$$ is a global defensive k-alliance in G if (i) each vertex not in S has a neighbor in S and (ii) each vertex of S has at least k more neighbors inside S than outside of it. The global defensive k-alliance number of G is the minimum cardinality among all global defensive k-alliance in G. In this paper this concept is studied on the generalized hierarchical, the lexicographic, the corona, and the edge corona product. For all of these products upper bounds expressed with related invariants of the factors are given. Sharpness of the bounds is also discussed.

Alliance polynomial of regular graphs

The alliance polynomial of a graph G with order n and maximum degree Δ is the polynomial A(G;x)=∑k=−ΔΔAk(G)xn+k, where Ak(G) is the number of exact defensive k-alliances in G. We obtain some properties of A(G;x) and its coefficients for regular graphs. In particular, we characterize the degree of regular graphs by the number of non-zero coefficients of their alliance polynomial. Besides, we prove that the family of alliance polynomials of Δ-regular graphs with small degree is a very special one, since it does not contain alliance polynomials of graphs which are not Δ-regular. By using this last result and direct computation we find that the alliance polynomial determines uniquely each cubic graph of order less than or equal to 10.

Distinctive power of the alliance polynomial for regular graphs

The alliance polynomial of a graph G with order n and maximum degree Δ is the polynomial A(G;x)=∑k=ΔΔAk(G)xn+k, where Ak(G) is the number of exact defensive k-alliances in G. We obtain some properties of A(G;x) and its coefficients for regular graphs. In particular, we characterize the degree of regular graphs by the number of non-zero coefficients of their alliance polynomial. Besides, we prove that the family of alliance polynomials of Δ-regular graphs with small degree is a very special one, since it does not contain alliance polynomials of graphs which are not Δ-regular.

Alliance free sets in Cartesian product graphs

Let G=(V,E) be a graph. For a non-empty subset of vertices S⊆V, and vertex v∈V, let δS(v)=|{u∈S:uv∈E}| denote the cardinality of the set of neighbors of v in S, and let S¯=V−S. Consider the following condition: (1)δS(v)≥δS¯(v)+k, which states that a vertex v has at least k more neighbors in S than it has in S¯. A set S⊆V that satisfies Condition (1) for every vertex v∈S is called a defensivek-alliance and for every vertex v in the open neighborhood of S is called an offensivek-alliance. A subset of vertices S⊆V is a powerfulk-alliance if it is both a defensive k-alliance and an offensive (k+2)-alliance. Moreover, a subset X⊂V is a defensive (an offensive or a powerful) k-alliance free set if X does not contain any defensive (offensive or powerful, respectively) k-alliance. In this article we study the relationships between defensive (offensive, powerful) k-alliance free sets in Cartesian product graphs and defensive (offensive, powerful) k-alliance free sets in the factor graphs.

Exact 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) ≥ d ¯ S (v )+ k. A defensive k-alliance S is called exact ,i fS 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. Furthermore, we characterize the exact defensive alliances in graph join G1 � G2 in terms of G1, G2. Mathematics Subject Classification: 05C69; 05A20; 05C50

Partitioning a Graph into Global Powerful k-Alliances

A set S of vertices of a graph is a defensive k-alliance if every vertex $${v\in S}$$ has at least k more neighbors in S than it has outside of S. Analogously, a set S is an offensive k-alliance if every vertex in the neighborhood of S has at least k more neighbors in S than it has outside of S. Also, a powerful k-alliance is a set S of vertices of the graph, which is both defensive k-alliance and offensive (k + 2)-alliance. A powerful k-alliance is called global if it is a dominating set. In this paper we show that for k ≥ 0, no graph is partitionable into global powerful k-alliances and, for k ≤ −1, we obtain upper bounds on the maximum number of sets belonging to a partition of a graph into global powerful k-alliances. In addition, we study the close relationships that exist between partitions of a Cartesian product graph, Γ1 × Γ2, into (global) powerful (k 1 + k 2)-alliances and partitions of Γi into (global) powerful k i -alliances, $${i\in \{1,2\}}$$.

Partitioning a graph into defensive k-alliances

A defensive k-alliance in a graph is a set S of vertices with the property that every vertex in S has at least k more neighbors in S than it has outside of S. A defensive k-alliance S is called global if it forms a dominating set. In this paper we study the problem of partitioning the vertex set of a graph into (global) defensive k-alliances. The (global) defensive k-alliance partition number of a graph Θ = (V, E), (ψkgd (Γ)) ψkd(Γ), is defined to be the maximum number of sets in a partition of V such that each set is a (global) defensive k-alliance. We obtain tight bounds on ψkd(Θ) and ψkgd (Γ) in terms of several parameters of the graph including the order, size, maximum and minimum degree, the algebraic connectivity and the isoperimetric number. Moreover, we study the close relationships that exist among partitions of Γ1 × Γ2 into (global) defensive (k1 + k2)-alliances and partitions of Γi into (global) defensive ki-alliances, i ∈ {1, 2}.