Bounded Tree-Width and LOGCFL

We show that (1) the recognition of tree-width bounded graphs and (2) the decidability of graph properties-which are defined by finite equivalence relations on h-sourced graphs-on tree-width bounded graphs belong to the complexity class LOGCFL. This is the lowest complexity class known for these problems. Our result complements the research in a series of papers by Arnborg, Bodlaender, Chandrasekharan, Courcelle, Hedetniemi, Lagergren, Proskurowski, Reed, Robertson, Seymour, Seese, and many others.     

Dead Ends and Detours En Route to Total Syntheses of the 1990s A list of abbreviations can be found at the end of the article

From the very beginning organic chemistry and total synthesis have been intimately joined. In fact, one of the first things that freshmen in organic chemistry learn is how to join two molecules together to obtain a more complex one. Of course they still have a long way to go to become fully mature synthetic chemists, but they must have the primary instinct to build molecules, as synthesis is the essence of organic chemistry. With the different points of view that actually coexist in the chemical community about the maturity of the science (art, or both) of organic synthesis, it is clear that nowadays we know how to make almost all of the most complex molecules ever isolated. The primary question is how easy is it to accomplish? For the readers of papers describing the total synthesis of either simple or complex molecules, it appears that the routes followed are, most of the time, smooth and free of troubles. The synthetic scheme written on paper is, apparently, done in the laboratory with few, if any, modifications and these, essentially, seem to be based on finding the optimal experimental conditions to effect the desired reaction. Failures in the planned synthetic scheme to achieve the goal, detours imposed by unexpected reactivity, or the absence of reactivity are almost never discussed, since they may diminish the value of the work reported. This review attempts to look at total synthesis from a different side; it will focus on troubles found during the synthetic work that cause detours from the original synthetic plan, or on the dead ends that eventually may force redesign. From there, the evolution from the original route to the final successful one that achieves the synthetic target will be presented. The syntheses discussed in this paper have been selected because they contain explicit information about the failures of the original synthetic plan, together with the evolution of the final route to the target molecule. Therefore, they contain a lot of useful negative information that may otherwise be lost.