  The present study was concerned with network failure problems for simple
connected undirected graphs. A connected graph becomes unconnected through edge
failure, under the assumptions that only edges can fail and each edge has an
identical failure distribution. The main purpose of the present study was to
show recurrent relations with respect to the number of edges in graph
generation procedures. To this end, simple connected undirected graphs were
correlated to random $K$-out-of-$N$ systems, and key features of such systems
were applied. In addition, some simple graph cases and examples were analyzed.
