We show that for each single crossing graph $H$, a polynomially bounded weight function for all $H$-minor free graphs $G$ can be constructed in Logspace such that it gives nonzero weights to all the cycles in $G$. This class of graphs subsumes almost all classes of graphs for which such a weight function is know to be constructed in Logspace. As a consequence, we obtain that for the class of $H$-minor free graphs where $H$ is a single crossing graph, reachability can be solved in UL, and bipartite maximum matching can be solved in SPL, which are small subclasses of the parallel complexity class NC. In the restrictive case of bipartite graphs, our maximum matching result improves upon the recent result of Eppstein and Vazirani, where they show an NC bound for constructing perfect matching in general single crossing minor free graphs.
翻译:我们显示,对于每个单交叉图,$H$,所有美元-美元-美元-美元-美元-美元-美元-美元-免费图形,在Logspace中可以构造一个多式捆绑的重量函数,使所有周期以$G值计算其非零加权值。这一类图形在Logspace中几乎包含所有类型的图表,而已知在Logspace中可以构造这种加权值。因此,我们获得的是,对于以$H美元-美元-美元-美元-美元-美元-美元-美元-免费图形为单一交叉图的类别,可以UL解决可达性,而双边的最大匹配值可以在SPL中解决,后者是平行复杂级NC的小型子类。在有限制性的双边图表中,我们的最大匹配结果随着最近Eppstein和Vazirani的结果而得到改善,其中显示NC将用来在普通单交叉免费图表中构建完美匹配。