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ON FINDING CRITICAL INDEPENDENT AND VERTEX SETS ALEXANDER A. AGEEV
 

Summary: ON FINDING CRITICAL INDEPENDENT AND VERTEX SETS
ALEXANDER A. AGEEV 
Abstract. An independent set Ic of a undirected graph G is called critical if
jI c j jN(Ic )j = maxfjIj jN(I)j : I is an independent set of Gg;
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 nding a
critical independent set is polynomially solvable. This paper shows that the problem can be solved
in O(jV (G)j 1=2 jE(G)j) time and its weighted version in O(jV (G)j 2 jE(G)j 1=2 ) time.
Key words. independent set, minimum cut
AMS subject classi cations. 05C35, 68R10
1. Introduction. Denote by G a simple undirected graph and by N(U ), U 
V (G), the set of all vertices of G adjacent to some vertex of U . An independent set
I c  V (G) is called critical if
(G) = jI c j jN(I c )j = maxfjI j jN(I)j : I is an independent set of Gg:
The number (G) is a parameter of G, closely related to some other important ones
[Zh90]. A vertex set U c is called critical if
(G) = jU c j jN(U c )j = maxfjU j jN(U)j : U  V (G)g:
Cun-Quan Zhang [Zh90] observed that (G) = (G) and that the problem of
nding critical independent set is reducible to the problem of nding critical vertex
set. In [Zh90] it is also shown by rather a sophisticated reduction to the linear

  

Source: Ageev, Alexandr - Sobolev Institute of Mathematics, Russian Academy of Sciences, Novosibirsk

 

Collections: Mathematics