How can I export the constraints of igraph to a data.frame but without including the node identifiers?

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You can use the as_data_frame() function from igraph and then remove the first column which contains the node identifiers. Here is an example:

library(igraph)

# create a sample graph with constraints
g <- graph.formula(1-2-3-4-5, 2-4-5)
set.vertex.attribute(g, "c1", c(1, 0, 0, 0, 1))
set.vertex.attribute(g, "c2", c(1, 0, 0, 1, 1))

# export constraints to data frame without node identifiers
df <- as_data_frame(g, what = "vertices", attrs = c("c1", "c2"))
df <- df[, -1]

The resulting data frame df will have the constraints in columns, without the node identifiers.