The n-dimensional hypercube Qn has many maximal independent sets of vertices. We study the cardinality of those maximal independent sets which are balanced , i.e. exactly half of whose vertices have even weight, obtaining both upper and lower bounds for the maximum value. For n ≤ 7 we obtain the exact value. For all odd n, we conjecture that the exact value is 2n−1 − ( n−1 (n−1)/2 ) . We also c...

The independent domination number of a graph G, denoted i(G), is the minimum cardinality of a maximal independent set of G. A maximal independent set of cardinality i(G) in G we call an i(G)-set. The graph G is called i-excellent if every vertex of G belongs to some i(G)-set. We provide a constructive characterization of i-excellent trees. c © 2002 Elsevier Science B.V. All rights reserved.

The trace of a set F on a another set X is F |X = F ∩ X and the trace of a family F of sets on X is FX = {F |X : F ∈ F}. In this note we prove that if a k-uniform family F ⊂ ([n] k ) has the property that for any k-subset X the trace F|X does not contain a maximal chain (a family C0 ⊂ C1 ⊂ ... ⊂ Ck with |Ci| = i), then |F| ≤ ( n−1 k−1 ) . This bound is sharp as shown by {F ∈ ([n] k ) , 1 ∈ F}. ...

Theaverage lower independencenumber iav(G)of a graphG=(V ,E) is defined as 1 |V | ∑ v∈V iv(G), and the average lower domination number av(G) is defined as 1 |V | ∑ v∈V v(G), where iv(G) (resp. v(G)) is the minimum cardinality of a maximal independent set (resp. dominating set) that contains v.We give an upper bound of iav(G) and av(G) for arbitrary graphs. Then we characterize the graphs achiev...

We study graphs in which the maximum and the minimum sizes of a maximal independent set differ by exactly one. We call these graphs almost well-covered, in analogy with the class of well-covered graphs, in which all maximal independent sets have the same size. A characterization of graphs of girth at least 8 having exactly two different sizes of maximal independent sets due to Finbow, Hartnell,...

Blbzsik, Z., M. Hujter, A. Pluhir and Z. Tuza, Graphs with no induced C4 and 2K,, Discrete Mathematics 115 (1993) 51-55. We characterize the structure of graphs containing neither the 4-cycle nor its complement as an induced subgraph. This self-complementary class B of graphs includes split graphs, which are graphs whose vertex set is the union of a clique and an independent set. In the members...

It is shown that in star-free graphs the maximum independent set problem, the minimum dominating set problem and the minimum independent dominating set problem are approximable up to constant factor by any maximal independent set.