Level of repair analysis and minimum cost homomorphisms of graphs
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Level of repair analysis and minimum cost homomorphisms of graphs
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Using critical sets to solve the maximum independent set problem
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An independent set $I_{c}$ of a undirected graph $G$ is called critical if \[|I_{c}|-|N(I_{c})|=\max\{|I|-|N(I)|:\mbox{\rm $I$ is an independent set of $G$}\},\] where $N(I)$ is the set of all vertices of $G$ adjacent to some vertex of $I$. It has been proved by Cun-Quan Zhang [SIAM J. Discrete Math., 3 (1990), pp. 431--438] that the problem of finding a critical independent set is polynomially solvable. This paper shows that the problem can be solved in $O(|V(G)|^{1/2}|E(G)|)$ time and its weighted version in $O(|V(G)|^{2}|E(G)|^{1/2})$ time.