A graph G is well-covered provided each maximal independent set of vertices has the same cardinality. The term sk of the independence sequence (s0, s1, . . . , sα) equals the number of independent k-sets of vertices of G. We investigate constraints on the linear orderings of the terms of the independence sequence of well-covered graphs. In particular, we provide a counterexample to the recent u...

A graph is well covered if every maximal independent set has the same size and very well covered if every maximal independent set contains exactly half the number of vertices. In this paper, we present an alternative characterization of a certain sub-class of well-covered graphs and show that this generalizes a characterization of very well covered graphs given by Favaron 3].

Given a system G = (G1,G2, . . . ,Gm) of m graphs on the same vertex set V , define the “joint independence number” α∩(G) as the maximal size of a set which is independent in all graphs Gi . Let also γ∪(G) be the “collective domination number” of the system, which is the minimal number of neighborhoods, each taken from any of the graphs Gi , whose union is V . König’s classical duality theorem ...

There are distributed graph algorithms for finding maximal matchings and independent sets in O ( Δ + log * n ) communication rounds; here, is the number of nodes maximum degree. The lower bound by Linial (1987, 1992) shows that dependency on optimal: These problems cannot be solved o (log rounds even if = 2. However, a long-standing open question, there currently an exponential gap between uppe...

In the early years of set theory, Du Bois Reymond introduced a vague notion of infinitary pantachie meant to symbolize an infinity bigger than the infinity of real numbers. Hausdorff reformulated this concept rigorously as a maximal chain (a linearly ordered subset) in a partially ordered set of certain type, for instance, the set N N under eventual domination. Hausdorff proved the existence of...

We present the first polynomial-time approximation scheme (PTAS) for the Minimum Independent Dominating Set problem in graphs of polynomially bounded growth. Graphs of bounded growth are used to characterize wireless communication networks, and this class of graph includes many models known from the literature, e.g. (Quasi) Unit Disk Graphs. An independent dominating set is a dominating set in ...

Min PB is the class of minimization problems whose objective functions are bounded by a polynomial in the size of the input. We show that there exist several problems that are Min PB-complete with respect to an approximation preserving reduction. These problems are very hard to approximate; in polynomial time they cannot be approximated within nε for some ε > 0, where n is the size of the input...