The critical independence number and an independence decomposition

An independent set Ic is a critical independent set if |Ic|−|N(Ic)| ≥ |J | − |N(J)|, for any independent set J . The critical independence number of a graph is the cardinality of a maximum critical independent set. This number is a lower bound for the independence number and can be computed in polynomial-time. Any graph can be decomposed into two subgraphs where the independence number of one subgraph equals its critical independence number, where the critical independence number of the other subgraph is zero, and where the sum of the independence numbers of the subgraphs is the independence number of the graph. A proof of a conjecture of Graffiti.pc yields a new characterization of König-Egervary graphs: these are exactly the graphs whose independence and critical independence numbers are equal.