  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 known 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.
