A note on the size of minimal covers

For the class of monotone boolean functions f : {0,1}n → {0,1} where all 1-certificates have size 2, we prove the tight bound n (λ+ 2)2/4, where λ is the size of the largest 0-certificate of f . This result can be translated to graph language as follows: for every graph G= (V ,E) the inequality |V | (λ+ 2)2/4 holds, where λ is the size of the largest minimal vertex cover of G. In addition, there are infinitely many graphs for which this inequality is tight. © 2006 Elsevier B.V. All rights reserved.