One can define the independence polynomial of a graph G as follows. Let ik(G) denote the number of independent sets of size k of G, where i0(G) = 1. Then the independence polynomial of G is

A transversal matroid M can be represented by a collection of sets, called a presentation of M , whose partial transversals are the independent sets of M . Minimal presentations are those for which removing any element from any set gives a presentation of a different matroid. We study the connections between (single-element) transversal extensions of M and extensions of presentations of M . We ...

A subset $S$ of vertex set $V(D)$ is an {em indpendent dominating set} of $D$ if $S$ is both an independent and a dominating set of $D$. The {em indpendent domination number}, $i(D)$ is the cardinality of the smallest independent dominating set of $D$. In this paper we calculate the independent domination number of the { em cartesian product} of two {em directed paths} $P_m$ and $P_n$ for arbi...

A dominating set S of a graph G is a global (strong) defensive alliance if for every vertex v ∈ S, the number of neighbors v has in S plus one is at least (greater than) the number of neighbors it has in V \ S. The dominating set S is a global (strong) offensive alliance if for every vertex v ∈ V \ S, the number of neighbors v has in S is at least (greater than) the number of neighbors it has i...

In every dense poset P every maximal antichain S may be partitioned into disjoint subsets S1 and S2 , such that the union of the upset of S1 with the downset of S2 yields the entire poset: U(S1) [ D(S2) = P . To nd a similar splitting of maximal antichains in posets is NP{hard in general.

A k-dominating set is a set D k V such that every vertex i 2 V nD k has at least k i neighbours in D k. The k-domination number k (G) of G is the cardinality of a smallest k-dominating set of G. For k 1 = ::: = k n = 1, k-domination corresponds to the usual concept of domination. Our approach yields an improvement of an upper bound for the domination number found then the conception of k-domina...

Let ` be a fixed rational prime number. Consider function fields K|k over algebraically closed fields k of characteristic 6= `. For each such a function field K|k, let Π K := Gal(K ′′|K) be the Galois group of a maximal pro-` abelian-by-central Galois extension K ′′|K, and ΠK = Gal(K ′|K) be the Galois group of the maximal pro-` abelian sub-extension K ′|K of K ′′|K. At the beginning of the 199...

A subset X of the vertex set of a graph G is a secure dominating set of G if X is a dominating set of G and if, for each vertex u not in X, there is a neighbouring vertex v of u in X such that the swap set (X − {v}) ∪ {u} is again a dominating set of G. The secure domination number of G is the cardinality of a smallest secure dominating set of G. A graph G is p-stable if the largest arbitrary s...

We present an O(1.1034) algorithm computing a maximum independent set of a graph with maximal degree 3. This result improves currently best upper bound of O(1.1255) for the problem obtained by Chen et al [2].

Let ir(G), γ(G), i(G), β0(G), Γ(G) and IR(G) be the irredundance number, the domination number, the independent domination number, the independence number, the upper domination number and the upper irredundance number of a graph G, respectively. In this paper we show that for any integers k1, k2, k3, k4, k5 there exists a cubic graph G satisfying the following conditions: γ(G)−ir(G) ≥ k1, i(G)−...