S. Klein, C.M.H. de Figueiredo, The NP-completeness of multi-partite cutset testing, Congr. Numer. 119 (1996) 217--222. Home/Search Document Details and Download
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Complexity of Graph Partition Problems - Tomas Feder Pavol (1999) (3 citations) Self-citation (Klein) (Correct)
.... three nonempty sets A;B;C such that B is independent and A;C are completely nonadjacent (in other words, no vertex of A is adjacent to any vertex of C) Tucker [22] proved that a minimum counterexample to the Strong Perfect Graph Conjecture does not admit a stable cutset; de Figueiredo and Klein [13] showed that recognizing graphs with a stable cutset is NP complete. A clique cutset similarly corresponds to a partition of the vertex set into three nonempty sets A;B, and C, such that B is complete and A and C are completely nonadjacent. Clique cutsets can be found in polynomial time, and lead ....
....problem is ffl NP complete when M has rows 011; 101; 110 (the 3 coloring problem) or 110; 011; 101 (the stable cutset problem) or 211; 121; 112 or 112; 211; 121 (their complements) and ffl polynomial time solvable otherwise. Proof. The NP complete cases are standard results [17, 13]. Consider now a matrix M with rows A;B;C and connections AB;AC;BC, which is different from the exceptions matrices listed in the theorem. We may assume that M is connected, thus at most one of the connections is 0. We may also assume that no row has both a 0 and a 2, cf. above, and, in ....
C. M. H. de Figueiredo and S. Klein. The NPcompleteness of multipartite cutset testing, Congressus Numerantium 119 (1996): 217--222.
List Partitions - Feder, Hell, Klein, Motwani (2003) Self-citation (Klein) (Correct)
....analogously (B is stable) and also corresponds 5 to an M partition with all parts non empty. Stable cutsets are of interest because a result of Tucker [36] asserts that a minimal imperfect graph other than an odd cycle cannot contain a stable cutset; this problem has been proved NP complete in [24]. The two clique cutset problem is defined similarly as a union of two complete subgraphs that disconnects the input graph. Its matrix is given in Figure 3, and we discuss it further in Section 5, where we give a subexponential algorithm for the problem. Whether or not it admits a polynomial time ....
....this for the special skew cutsets where B consists of a single vertex; for this case he also gave a polynomial time recognition algorithm. The conjecture has also been established [12] if it is required that B and D are both stable (B = D = 0) in this case the recognition problem is NP complete [24]. Chv atal posed the problem of the complexity of finding a general skew cutset. In [22] we offered the first subexponential time algorithm, strongly suggesting that the problem is not NP complete. Most recently, one of us (Klein) together with de Figueiredo, Kohayakawa, and Reed, indeed found ....
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C. M. H. de Figueiredo and S. Klein, The NP-completeness of multipartite cutset testing, Congressus Numerantium 119 (1996) 217--222.
On Stable Cutsets in Graphs - Brandstädt, Dragan, Le, Szymczak (2000) (8 citations) (Correct)
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S. Klein, C.M.H. de Figueiredo, The NP-completeness of multi-partite cutset testing, Congr. Numer. 119 (1996) 217--222.
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