Q:

trace connections in network?

How can I find/isolate all the upstream/downstream connections to a node in a directed graph?
For instance, in R igraph I create two paths A->B->C and D->B->E:
library(igraph)

g <- graph_from_literal(A-+B-+C, D-+B-+E)

plot(g, vertex.color = "grey", edge.color = "blue")

By selecting the node C or A, I would like to retrieve A->B->C and D->B->C. What is this operation called? Is it possible to invoke this functionality through R/igraph?

A:

This completes the answer started by @Psidom. 
You seem to be looking only for maximal paths to or from a node. A maximal path out of a node ends in a sink. A maximal path into a node begins at a source. I provide slightly different solutions for the paths into or out of a node. 
Using your sample data:
sources = which(degree(g, v = V(g), mode = "in")==0,  useNames = T)
sinks   = which(degree(g, v = V(g), mode = "out")==0, useNames = T)

## Paths out of A
all_simple_paths(g, from = "A", to = sinks)
[[1]]
+ 3/5 vertices, named:
[1] A B C

[[2]]
+ 3/5 vertices, named:
[1] A B E

## Paths into C
lapply(sources, function(x) all_simple_paths(g, from =x, to = "C"))
$A
$A[[1]]
+ 3/5 vertices, named:
[1] A B C

$D
$D[[1]]
+ 3/5 vertices, named:
[1] D B C

