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

Usage

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.

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 058277c:
#> [1]  7 37 55 56
#> 
#> [[2]]
#> + 4/100 vertices, from 058277c:
#> [1]  7 55 56 69
#> 
#> [[3]]
#> + 4/100 vertices, from 058277c:
#> [1]  7 56 69 74
#> 
#> [[4]]
#> + 4/100 vertices, from 058277c:
#> [1]  8 15 73 80
#> 
#> [[5]]
#> + 4/100 vertices, from 058277c:
#> [1]  8 15 73 84
#> 
#> [[6]]
#> + 4/100 vertices, from 058277c:
#> [1] 13 16 37 40
#> 
#> [[7]]
#> + 4/100 vertices, from 058277c:
#> [1] 21 32 45 61
#> 
#> [[8]]
#> + 4/100 vertices, from 058277c:
#> [1] 22 55 56 64
#> 
#> [[9]]
#> + 4/100 vertices, from 058277c:
#> [1] 23 69 75 90
#> 
largest_ivs(g)
#> [[1]]
#> + 4/100 vertices, from 058277c:
#> [1] 21 32 45 61
#> 
#> [[2]]
#> + 4/100 vertices, from 058277c:
#> [1]  7 37 55 56
#> 
#> [[3]]
#> + 4/100 vertices, from 058277c:
#> [1]  7 55 56 69
#> 
#> [[4]]
#> + 4/100 vertices, from 058277c:
#> [1]  7 56 69 74
#> 
#> [[5]]
#> + 4/100 vertices, from 058277c:
#> [1]  8 15 73 80
#> 
#> [[6]]
#> + 4/100 vertices, from 058277c:
#> [1]  8 15 73 84
#> 
#> [[7]]
#> + 4/100 vertices, from 058277c:
#> [1] 22 55 56 64
#> 
#> [[8]]
#> + 4/100 vertices, from 058277c:
#> [1] 23 69 75 90
#> 
#> [[9]]
#> + 4/100 vertices, from 058277c:
#> [1] 13 16 37 40
#> 
# Empty graph
induced_subgraph(g, largest_ivs(g)[[1]])
#> IGRAPH 147361e U--- 4 0 -- Erdos-Renyi (gnp) graph
#> + attr: name (g/c), type (g/c), loops (g/l), p (g/n)
#> + edges from 147361e:

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