Independent vertex sets

A vertex set is called independent if there no edges between any two vertices in it. These functions find independent vertex sets in undirected graphs

ivs(graph, min = NULL, max = NULL)

largest_ivs(graph)

max_ivs(graph)

ivs_size(graph)

independence_number(graph)

Arguments

graph

The input graph, directed graphs are considered as undirected, loop edges and multiple edges are ignored.

min

Numeric constant, limit for the minimum size of the independent vertex sets to find. NULL means no limit.

max

Numeric constant, limit for the maximum size of the independent vertex sets to find. NULL means no limit.

Value

ivs(), largest_ivs() and max_ivs() return a list containing numeric vertex ids, each list element is an independent vertex set.

ivs_size() returns an integer constant.

Details

ivs() finds all independent vertex sets in the network, obeying the size limitations given in the min and max arguments.

largest_ivs() finds the largest independent vertex sets in the graph. An independent vertex set is largest if there is no independent vertex set with more vertices.

max_ivs() finds the maximal independent vertex sets in the graph. An independent vertex set is maximal if it cannot be extended to a larger independent vertex set. The largest independent vertex sets are maximal, but the opposite is not always true.

ivs_size() calculate the size of the largest independent vertex set(s).

independence_number() is an alias for ivs_size().

These functions use the algorithm described by Tsukiyama et al., see reference below.

References

S. Tsukiyama, M. Ide, H. Ariyoshi and I. Shirawaka. A new algorithm for generating all the maximal independent sets. SIAM J Computing, 6:505–517, 1977.

See also

Other cliques: cliques(), weighted_cliques()

Author

Tamas Nepusz ntamas@gmail.com ported it from the Very Nauty Graph Library by Keith Briggs (http://keithbriggs.info/) and Gabor Csardi csardi.gabor@gmail.com wrote the R interface and this manual page.

Examples

# Do not run, takes a couple of seconds

# A quite dense graph
set.seed(42)
g <- sample_gnp(100, 0.9)
ivs_size(g)
#> [1] 4
ivs(g, min = ivs_size(g))
#> [[1]]
#> + 4/100 vertices, from db4d191:
#> [1]  7 37 55 56
#> 
#> [[2]]
#> + 4/100 vertices, from db4d191:
#> [1]  7 55 56 69
#> 
#> [[3]]
#> + 4/100 vertices, from db4d191:
#> [1]  7 56 69 74
#> 
#> [[4]]
#> + 4/100 vertices, from db4d191:
#> [1]  8 15 73 80
#> 
#> [[5]]
#> + 4/100 vertices, from db4d191:
#> [1]  8 15 73 84
#> 
#> [[6]]
#> + 4/100 vertices, from db4d191:
#> [1] 13 16 37 40
#> 
#> [[7]]
#> + 4/100 vertices, from db4d191:
#> [1] 21 32 45 61
#> 
#> [[8]]
#> + 4/100 vertices, from db4d191:
#> [1] 22 55 56 64
#> 
#> [[9]]
#> + 4/100 vertices, from db4d191:
#> [1] 23 69 75 90
#> 
largest_ivs(g)
#> [[1]]
#> + 4/100 vertices, from db4d191:
#> [1] 21 32 45 61
#> 
#> [[2]]
#> + 4/100 vertices, from db4d191:
#> [1]  7 37 55 56
#> 
#> [[3]]
#> + 4/100 vertices, from db4d191:
#> [1]  7 55 56 69
#> 
#> [[4]]
#> + 4/100 vertices, from db4d191:
#> [1]  7 56 69 74
#> 
#> [[5]]
#> + 4/100 vertices, from db4d191:
#> [1]  8 15 73 80
#> 
#> [[6]]
#> + 4/100 vertices, from db4d191:
#> [1]  8 15 73 84
#> 
#> [[7]]
#> + 4/100 vertices, from db4d191:
#> [1] 22 55 56 64
#> 
#> [[8]]
#> + 4/100 vertices, from db4d191:
#> [1] 23 69 75 90
#> 
#> [[9]]
#> + 4/100 vertices, from db4d191:
#> [1] 13 16 37 40
#> 
# Empty graph
induced_subgraph(g, largest_ivs(g)[[1]])
#> IGRAPH c244d66 U--- 4 0 -- Erdos-Renyi (gnp) graph
#> + attr: name (g/c), type (g/c), loops (g/l), p (g/n)
#> + edges from c244d66:

length(max_ivs(g))
#> [1] 326