  This paper continues the investigation of the configuration space of two
distinct points on a graph. We analyze the process of adding an additional edge
to the graph and the resulting changes in the topology of the configuration
space. We introduce a linking bilinear form on the homology group of the graph
with values in the cokernel of the intersection form (introduced in Part I of
this work). For a large class of graphs, which we call mature graphs, we give
explicit expressions for the homology groups of the configuration space. We
show that under a simple condition, adding an edge to a mature graph yields
another mature graph.
