  We consider changes in properties of a subgraph of an infinite graph
resulting from the addition of open edges of Bernoulli percolation on the
infinite graph to the subgraph. We give the triplet of an infinite graph, one
of its subgraphs, and a property of the subgraphs. Then, in a manner similar to
the way Hammersley's critical probability is defined, we can define two values
associated with the triplet. We regard the two values as certain critical
probabilities, and compare them with Hammersley's critical probability. In this
paper, we focus on the following cases of a graph property: being a transient
subgraph, having finitely many cut points or no cut points, being a recurrent
subset, or being connected. Our results depend heavily on the choice of the
triplet.
  Most results of this paper are announced in \cite{O16} without proofs. This
paper gives full details of them.
