Vertex packings: Structural properties and algorithms

We consider a binary integer programming formulation (VP) for the weighted vertex packing problem in a simple graph. A sufficient “local” optimality condition for (VP) is given and this result is used to derive relations between (VP) and the linear program (VLP) obtained by deleting the integrality restrictions in (VP). Our most striking result is that those variables which assume binary values in an optimum (VLP) solution retain the same values in an optimum (VP) solution. This result is of interest because variables are (0, 1/2, 1). valued in basic feasible solutions to (VLP) and (VLP) can be solved by a “good” algorithm. This relationship and other optimality conditions are incorporated into an implicit enumeration algorithm for solving (VP). Some computational experience is reported.     

POCl3-mediated synthesis of hydrolysis-prone 2-trifluoroethylbenzimidazoles

Loss of a trifluoromethyl group has been observed as an unproductive side reaction in the formation of 2-trifluoroethylbenzimidazoles. A hydrolytic mechanism for this transformation is proposed that is consistent with evidence for the identity of reaction intermediates. Cyclodehydration with POCl₃ suppresses hydrolysis of the trifluoromethyl group and provides 2-trifluoroethylbenzimidazoles in good yields.     

Embedding finite posets in cubes

In this paper we define the n-cube Q_n as the poset obtained by taking the cartesian product of n chains each consisting of two points. For a finite poset X, we then define dim₂X as the smallest positive integer n such that X can be embedded as a subposet of Q_n. For any poset X we then have log₂ |X| ? dim₂X ? |X|. For the distributive lattice $\text{L =}\text{2}{}^{\mathrm{X}}$, dim₂L = |X| and for the crown S^k_n, dim₂ (S^k_n) = n + k. For each k ? 2, there exist positive constants c₁ and c₂ so that for the poset X consisting of all one element and k-element subsets of an n-element set, the inequality c₁ log₂n < dim₂(X) < c₂ log₂n holds for all n with k < n. A poset is called Q-critical if dim₂ (X ? x) < dim₂(X) for every x ? X. We define a join operation ⊕ on posets under which the collection Q of all Q-critical posets which are not chains forms a semigroup in which unique factorization holds. We then completely determine the subcollection $\text{M}$ ? Q consisting of all posets X for which dim₂ (X) = |X|.