Graph theoretic independence and critical independent sets

The independence number of a graph is the maximum number of pairwise nonadjacent vertices of the graph. New bounds are presented for this NP-hard invariant, new algorithms for calculating it, as well as new theoretical techniques for investigating maximum independent sets in a graph. These include: 1. A new, short proof of Graffiti's conjectured lower bound for the independence number in terms of the number of cut vertices of the graph. 2. A new proof of an upper bound for the independence number in terms of the number of cut vertices, and (together with G. Henry, R. Pepper, and D. Sexton) the characterization of those graphs where equality of this bound holds. 3. A faster version of Tarjan and Trojanowski's algorithm for finding maximum independent sets in fullerenes, together with a previously unreported computation adding to the evidence that minimizing independence is the best statistical predictor of fullerene stability. 4. A characterization of those graphs whose independence number equals its radius, which was an open problem mentioned in a 1986 paper of Fajtlowicz and Waller. 5. A new sufficient condition, following a conjecture of Graffiti, for the existence of a Hamiltonian cycle in a graph. 6. A characterization of those graphs whose independence number equals its annihilation number, which was an open problem in Pepper's 2004 dissertation 7. A polynomial-time algorithm for finding maximum cardinality critical independent sets, which was an open problem of Butenko and Trukhanov's 2005 preprint vii on using critical independent sets in order to speed-up finding maximum independent sets. 8. The invention of the critical independence number, a new polynomial-time computable lower-bound for the independence number, and a polynomial-time computable characterization of the graphs where these invariants are equal. 9. A decomposition of a general graph into two unique subgraphs, such that the independence numbers of these graphs are additive, and where one of these independence number computations can be done in polynomial-time. 10. A new and simple characterization of Konig-Egervary graphs, resulting from a surprising conjecture of Graffiti.pc.

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