Aqua one-pot,three-component synthesis of dihydropyrano[3,2-c]chromenes and amino-benzochromenes catalyzed by sodium malonate

Research on Chemical Intermediates - The widespread biological properties of dihydropyrano[3,2-c]chromenes, 2-amino-4-aryl-4H-benzo[h]chromenes, and 3-amino-1-aryl-1H-benzo[f]chromenes have led to...     

Shortest paths in linear time on minor-closed graph classes, with an application to Steiner tree approximation

We generalize the linear-time shortest-paths algorithm for planar graphs with nonnegative edge-weights of Henzinger et al. (1994) to work for any proper minor-closed class of graphs. We argue that their algorithm can not be adapted by standard methods to all proper minor-closed classes. By using recent deep results in graph minor theory, we show how to construct an appropriate recursive division in linear time for any graph excluding a fixed minor and how to transform the graph and its division afterwards, so that it has maximum degree three. Based on such a division, the original framework of Henzinger et al. can be applied. Afterwards, we show that using this algorithm, one can implement Mehlhorn’s (1988) 2-approximation algorithm for the Steiner tree problem in linear time on these graph classes.