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NBMOs of non-disjoint (top) and disjoint (bottom) Non-Kekulé molecules Non-Kekulé molecules with two formal radical centers (non-Kekulé diradicals) can be classified into non-disjoint and disjoint by the shape of their two non-bonding molecular orbitals (NBMOs). Both NBMOs of molecules with non-disjoint characteristics such as trimethylenemethane have electron density at the same atom. According to Hund's rule, each orbital is filled with one electron with parallel spin, avoiding the Coulomb repulsion by filling one orbital with two electrons. Therefore, such molecules with non-disjoint NBMOs are expected to prefer a triplet ground state.
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This is because it cannot be written as a union of two disjoint open sets, but it can be written as a union of two (non-disjoint) closed sets.
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Descriptively close sets contain elements that have matching descriptions. Such sets can be either disjoint or non-disjoint sets. Spatially near sets are also descriptively near sets. Figure 2.
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In contrast, the NBMOs of the molecules with disjoint characteristics such as tetramethyleneethane can be described without having electron density at the same atom. With such MOs, the destabilization factor by the Coulomb repulsion becomes much smaller than with non-disjoint type molecules, and therefore the relative stability of the singlet ground state to the triplet ground state will be nearly equal, or even reversed because of exchange interaction.
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The domination problem was studied from the 1950s onwards, but the rate of research on domination significantly increased in the mid-1970s. In 1972, Richard Karp proved the set cover problem to be NP-complete. This had immediate implications for the dominating set problem, as there are straightforward vertex to set and edge to non-disjoint-intersection bijections between the two problems. This proved the dominating set problem to be NP- complete as well.
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