We consider the problem of approximating the size of a minimum non-extendible independent set of a graph, also known as the minimum dominating independence number. We strengthen a result of Irving [2] to show that there is no constant > 0 for which this problem can be approximated within a factor of n 10 in polynomial time, unless P = NP. This is the strongest lower bound we are aware of for po...

In this paper we consider graphs which have no k vertex-disjoint cycles. For given integers k, let f (k, ) be the maximum order of a graph G with independence number (G) , which has no k vertex-disjoint cycles. We prove that f (k, ) = 3k + 2 − 3 if 1 5 or 1 k 2, and f (k, ) 3k + 2 − 3 in general. We also prove the following results: (1) there exists a constant c (depending only on ) such that f...

A partial Steiner (n, r, l)-system is an r-uniform hypergraph on n vertices in which every set of l vertices is contained in at most one edge. A partial Steiner (n, r, l)-system is complete if every set of l vertices is contained in exactly one edge. In a hypergraph H, the independence number α(H) denotes the maximum size of a set of vertices in H containing no edge. In this article we prove th...

For an integer k ≥ 1 and a graph G = (V,E), a subset S of V is kindependent if every vertex in S has at most k − 1 neighbors in S. The k-independent number βk(G) is the maximum cardinality of a kindependent set of G. In this work, we study relations between βk(G), βj(G) and the domination number γ(G) in a graph G where 1 ≤ j < k. Also we give some characterizations of extremal graphs.

There are many functions of the degree sequence of a graph which give lower bounds on the independence number of the graph. In particular, for every graph G, α(G) ≥ R(d(G)), where R is the residue of the degree sequence of G. We consider the precision of this estimate when it is applied to semi-regular degree sequences. We show that the residue nearly always gives the best possible estimate on ...

A lower bound for the independence number of a graph is a graph invariant l such that, for every graph G, l(G) ≤ α(G). Similarly, an upper bound for the independence number is a graph invariant u such that, for every graph G, α(G) ≤ u(G). Many efficiently computable upper and lower bounds, called α-bounds here, have been published and these are surveyed in the following section. They can be use...

Let G be a simple undirected graph. A broadcast on G is a function f : V (G) → N such that f(v) ≤ eG(v) holds for every vertex v of G, where eG(v) denotes the eccentricity of v in G, that is, the maximum distance from v to any other vertex of G. The cost of f is the value cost(f) = ∑ v∈V (G) f(v). A broadcast f on G is independent if for every two distinct vertices u and v in G, dG(u, v) > max{...

The concept of inverse domination was introduced by Kulli V.R. and Sigarakanti S.C. [9] . Let D be a  set of G. A dominating set D1  VD is called an inverse dominating set of G with respect to D. The inverse domination number   (G) is the order of a smallest inverse dominating set. Motivated by this definition we define another parameter as follows. Let D be a maximum independent set in G. ...

Let K (3) 4 denote the complete 3-uniform hypergraph on 4 vertices. Ajtai, Erdős, Komlós, and Szemerédi (1981) asked if there is a function ω(d) → ∞ such that every 3-uniform, K (3) 4 -free hypergraph H with N vertices and average degree d has independence number at least N d1/2 ω(d). We answer this question by constructing a 3-uniform, K (3) 4 -free hypergraph with independence number at most ...