Finding critical independent sets and critical vertex subsets are polynomial problems
SIAM Journal on Discrete Mathematics
On Finding Critical Independent and Vertex Sets
SIAM Journal on Discrete Mathematics
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Vertices Belonging to All Critical Sets of a Graph
SIAM Journal on Discrete Mathematics
Note: On maximum matchings in König-Egerváry graphs
Discrete Applied Mathematics
Forbidden subgraphs and the König-Egerváry property
Discrete Applied Mathematics
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An independent set I"c is a critical independent set if |I"c|-|N(I"c)|=|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 efficiently 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 Konig-Egervary graphs: these are exactly the graphs whose independence and critical independence numbers are equal.