### Table 1: NP-completeness cut-off points for a3 a20a46a32 a11 a60a32a17a12a58a6 -a15 a50 a57a10a61 a45a35a61 a45 a55 a47

"... In PAGE 13: ...Table 1: NP-completeness cut-off points for a3 a20a46a32 a11 a60a32a17a12a58a6 -a15 a50 a57a10a61 a45a35a61 a45 a55 a47 For example, a3 a52a10a32 a14 a32 a14 a64 a6 -a15 a50a41a57a16a61 a45a62a61 a45a41a55a35a47 is nothing else than the NP-complete domatic number problem: Given a graph a0 , decide whether or not a0 can be partitioned into three dominating sets. In contrast, a3 a12 a10a32 a14 a32 a14 a64 a6 -a15a58a50a41a57a16a61 a45a35a61 a45a41a55a35a47 is in P, and therefore the corresponding entry in Table1 is a52 for a11 a26 a14 and a12 a4a26 a14 a64 . A value of a3 in Table 1 means that this problem is efficiently solvable for all values of a20 .... In PAGE 13: ... In contrast, a3 a12 a10a32 a14 a32 a14 a64 a6 -a15a58a50a41a57a16a61 a45a35a61 a45a41a55a35a47 is in P, and therefore the corresponding entry in Table 1 is a52 for a11 a26 a14 and a12 a4a26 a14 a64 . A value of a3 in Table1 means that this problem is efficiently solvable for all values of a20 . The value of a12 a26 a27 a12a42a10a40 is not considered, since all graphs have a a3 a20a46a32 a11 a60a32 a27 a12a42a10a40a62a6 -partition if and only if they have the trivial partition into a20 disjoint a3 a11 a60a32 a27 a12a42a10a40a62a6 -sets a2 a30a7a26 a2 a3a5a0a7a6 and a2 a3 a26 a6 , for each a22 a18 a27 a1a12a10a32a37a36a37a36a37a36a38a32 a20 a40 .... In PAGE 13: ... The value of a12 a26 a27 a12a42a10a40 is not considered, since all graphs have a a3 a20a46a32 a11 a60a32 a27 a12a42a10a40a62a6 -partition if and only if they have the trivial partition into a20 disjoint a3 a11 a60a32 a27 a12a42a10a40a62a6 -sets a2 a30a7a26 a2 a3a5a0a7a6 and a2 a3 a26 a6 , for each a22 a18 a27 a1a12a10a32a37a36a37a36a37a36a38a32 a20 a40 . The problems in Table1 that are marked by a a19 are maximum problems, and the problems that are marked by a a23 are minimum problems. That is, for all a20 a18 a0 , we have that a3 a20 a19 a3a0a35a32 a11 a60a32a17a12a58a6 -a15 a50 a57a10a61 a45a35a61 a45 a55 a47 a21 a3 a20a46a32 a11 a60a32a17a12a58a6 -a15 a50 a57a10a61 a45a62a61 a45 a55 a47 for the maximum problems, and we have that a3 a20a46a32 a11 a60a32a17a12a14a6 -a15a58a50a41a57a16a61 a45a35a61 a45a41a55a35a47a62a21 a3 a20 a19 a0 a32 a11 a60a32a17a12a58a6 -a15 a50 a57a10a61 a45a35a61 a45 a55 a47 for the minimum problems.... In PAGE 14: ... To prove that the a3 a20a46a32 a11 a60a32a17a12a14a6 -a15a58a50a41a57a10a61 a45a35a61 a45a41a55 a47 problems with a12 a26 a9a14a68a64 are maximum problems, note that once we have found a a3 a20 a19 a3a0 a32 a11 a60a32 a14 a64 a6 -partition into a20 a68a19 a0 pairwise disjoint sets a2 a30a29a32 a2 a33a62a32a37a36a37a36a37a36a38a32 a2 a24 a64 a30 , the sets a2 a30a37a32 a2 a33a35a32a37a36a37a36a37a36a38a32 a2 a24 a0 a30 a32 a0 a2 a24 with a0 a2 a24 a26 a2 a24 a5 a2 a24 a64 a30 are a a3 a20a46a32 a11 a60a32 a14 a64 a6 -partition as well. Observe that those problems in Table1 that are marked neither by a a19 nor by a a23 are neither maximum nor minimum problems in the sense defined above. That is, we have neither a3 a20 a19 a0 a32 a11 a60a32a17a12a14a6 -a15 a50 a57a10a61 a45a62a61 a45 a55 a47 a21 a3 a20a46a32 a11 a60a32a17a12a14a6 -a15 a50a41a57a16a61 a45a62a61 a45a41a55a35a47 nor a3 a20a46a32 a11 a60a32a17a12a58a6 -a15 a50 a57a10a61 a45a62a61 a45 a55 a47 a21 a3 a20 a19 a3a0a35a32 a11 a60a32a17a12a58a6 -a15 a50 a57a10a61 a45a35a61 a45 a55 a47 , since for each a20 a18 a0 , there exist graphs a0 with a0 a10a18 a3 a20a46a32 a11 a60a32a17a12a58a6 -a15 a50 a57a10a61 a45a35a61 a45 a55 a47 and a0 a25 a18 a3 a1 a32 a11 a60a32a17a12a14a6 -a15a58a50a41a57a10a61 a45a35a61 a45a41a55 a47 for any a1 a18 a0 with a1 a25 a26 a20 .... In PAGE 19: ...3 The case a20a22a21a24a23a1a0 We consider the cases a11 a26 a14 and a11 a26 a14 a64 only. These two are the only maximum problems in Table1 . Recall that since a3 a20a46a32 a14 a32 a14 a64 a6 -a15a58a50a41a57a10a61 a45a35a61 a45a41a55 a47 and a3 a20a46a32 a14 a64 a32 a14 a64 a6 -a15a58a50a41a57a16a61 a45a35a61 a45a41a55a35a47 are maximum problems, their exact versions are defined as follows: a16a18a17 a50a14a54a35a61 -a3 a20a46a32 a11 a60a32 a14 a68a64 a6 -a15 a50a41a57a16a61 a45a62a61 a45a41a55a35a47 a26 a2 a0 a0 a10a18a25a3 a20a46a32 a11 a60a32 a14 a64 a6 -a15a58a50a41a57a10a61 a45a35a61 a45a41a55 a47 and a0 a25 a18 a25a3 a20 a19 a3a0a35a32 a11 a60a32 a14 a64 a6 -a15 a50 a57a10a61 a45a62a61 a45 a55 a47 a4a3 a32 where a11 a18 a27 a20a14a19a32 a14 a64 a40 .... ..."

Cited by 1

### Table 1: Operators in the proof of NP-completeness

1992

Cited by 14

### Table 1: Operators in the proof of NP-completeness

1992

Cited by 14

### Table 2: Review of NP-completeness results for graph layout problems.

2002

"... In PAGE 6: ... It is possible to show that many layout problems remain NP-complete even for cer- tain restricted classes of graphs. Table2 summarizes these known negative results and also includes references for the proofs of Theorem 1. Fixed parameter results.... ..."

Cited by 11

### Table 1. Summary of results. NPCmeans that the problem is NP-complete.

1997

"... In PAGE 3: ... Since we can prove that it is NP-complete to decide whether a 2-connected planar graph of maximum degree 2 T ? 1 has a T -spanning tree, this result establishes a complete characterization of the T -spanning tree problem for k-connected planar graphs of maximum degree G. Table1 summarizes the results (it assumes that G gt; T 2). Organization of the paper Section 2 provides basic terminology.... ..."

Cited by 5

### Table 1. Summary of results. NPCmeans that the problem is NP-complete.

1997

"... In PAGE 3: ... Since we can prove that it is NP-complete to decide whether a 2-connected planar graph of maximum degree 2 T ? 1 has a T -spanning tree, this result establishes a complete characterization of the T -spanning tree problem for k-connected planar graphs of maximum degree G. Table1 summarizes the results (it assumes that G gt; T 2). Organization of the paper Section 2 provides basic terminology.... ..."

Cited by 5

### Table 1. Summary of results. NPCmeans that the problem is NP-complete.

1997

"... In PAGE 3: ... Since we can prove that it is NP-complete to decide whether a 2-connected planar graph of maximum degree 2 T ? 1 has a T -spanning tree, this result establishes a complete characterization of the T -spanning tree problem for k-connected planar graphs of maximum degree G. Table1 summarizes the results (it assumes that G gt; T 2). Organization of the paper Section 2 provides basic terminology.... ..."

Cited by 5

### Table 2: Complexity of AHNEPs solving NP-complete problems

### Table 2: Possible Multiobjective NP-Complete Functions NP-Complete Problem Examples

"... In PAGE 4: ... This may prevent general comparison between various MOEAs, but the problems apos; inherent di culty should present the desired algorithmic challenges and complement numeric test suite MOPs. Table2 suggests possible NP- Complete MOPs for inclusion. To date, only two non-nu- merical MOP examples are found in the MOEA literature: one is a multiobjective NP-Complete example (a multiob- 0 0.... ..."