How to identify fully connected node clusters with igraph?

Question

I am trying to calculate network clusters using igraph in R, where all nodes are connected. The plot seems to be working fine, but then I cannot return the correct groupings from my clusters.

In this example, the graph shows 4 main clusters, but not all nodes are connected in the largest cluster:

I would like to be able to return the following list of clusters from this object graph:

[[1]]
[1] 8 9

[[2]]
[1]  7 10

[[3]]
[1]  4  6 11

[[4]]
[1] 2 3 5

[[5]]
[1]  1  3  5 12

Code example:

library(igraph)

topology <- structure(list(N1 = c(1, 3, 5, 12, 2, 3, 5, 1, 2, 3, 5, 12, 4, 
6, 11, 1, 2, 3, 5, 12, 4, 6, 11, 7, 10, 8, 9, 8, 9, 7, 10, 4, 
6, 11, 1, 3, 5, 12), N2 = c(1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 
3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 
11, 11, 11, 12, 12, 12, 12)), .Names = c("N1", "N2"), row.names = c(NA, 
-38L), class = "data.frame")

g2 <- graph.data.frame(topology, directed=FALSE)
g3 <- simplify(g2)
plot(g3)

The function cliquesgives me part of the path:

tmp <- cliques(g3)
tmp

but this list also gives groups in which not all nodes are connected. For example, this click includes nodes 1,2,3,5, but 1 connects only to 3, and 2 connects only to 3 and 5, and 5 connects only to 2:

topology[tmp[[31]],]
#  N1 N2
#6  3  2
#7  5  2
#8  1  3

Thanks in advance for your help.

Answer 1

maximal.cliques igraph. . .

# Load package
library(igraph)

# Load data
topology <- structure(list(N1 = c(1, 3, 5, 12, 2, 3, 5, 1, 2, 3, 5, 12, 4, 
6, 11, 1, 2, 3, 5, 12, 4, 6, 11, 7, 10, 8, 9, 8, 9, 7, 10, 4, 
6, 11, 1, 3, 5, 12), N2 = c(1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 
3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 
11, 11, 11, 12, 12, 12, 12)), .Names = c("N1", "N2"), row.names = c(NA, 
-38L), class = "data.frame")

# Get rid of loops and ensure right naming of vertices
g3 <- simplify(graph.data.frame(topology[order(topology[[1]]),],directed = FALSE))

# Plot graph
plot(g3)

# Calcuate the maximal cliques
maximal.cliques(g3)

# > maximal.cliques(g3)
# [[1]]
# [1] 9 8
# 
# [[2]]
# [1] 10  7
# 
# [[3]]
# [1] 2 3 5
# 
# [[4]]
# [1]  6  4 11
# 
# [[5]]
# [1] 12  1  5  3