Create a group index for values connected directly and indirectly

Using igraph get membership, then map on names:

library(igraph)

# convert to graph, and get clusters membership ids
g <- graph_from_data_frame(df1[, c(2, 3, 1)])
myGroups <- components(g)$membership

myGroups 
# A B C D E F Z X Y W V U s T 
# 1 1 2 3 4 4 1 1 1 2 2 2 3 4 

# then map on names
df1$group <- myGroups[df1$G1]


df1
#    id G1 G2 group
# 1   1  A  Z     1
# 2   2  A  X     1
# 3   3  B  X     1
# 4   4  B  Y     1
# 5   5  C  W     2
# 6   6  C  V     2
# 7   7  C  U     2
# 8   8  D  s     3
# 9   9  E  T     4
# 10 10  F  T     4

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