Macaulay2 » Documentation
Packages » EdgeIdeals :: allOddHoles
next | previous | forward | backward | up | index | toc

allOddHoles -- returns all odd holes in a graph

Synopsis

Description

An odd hole is an odd induced cycle of length at least 5. The method is based on work of Francisco-Ha-Van Tuyl, looking at the associated primes of the square of the Alexander dual of the edge ideal.

See C.A. Francisco, H.T. Ha, A. Van Tuyl, "Algebraic methods for detecting odd holes in a graph." (2008) Preprint. arXiv:0806.1159v1.

i1 : R = QQ[x_1..x_6];
 
i2 : G = graph({x_1*x_2,x_2*x_3,x_3*x_4,x_4*x_5,x_1*x_5,x_1*x_6,x_5*x_6}) --5-cycle and a triangle

o2 = Graph{"edges" => {{x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }}}
                         1   2     2   3     3   4     1   5     4   5     1   6     5   6
           "ring" => R
           "vertices" => {x , x , x , x , x , x }
                           1   2   3   4   5   6

o2 : Graph
 
i3 : allOddHoles G --only the 5-cycle should appear

o3 = {{x , x , x , x , x }}
        1   2   3   4   5

o3 : List
 
i4 : H = graph({x_1*x_2,x_2*x_3,x_3*x_4,x_4*x_5,x_1*x_5,x_1*x_6,x_5*x_6,x_1*x_4}) --no odd holes

o4 = Graph{"edges" => {{x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }}}
                         1   2     2   3     1   4     3   4     1   5     4   5     1   6     5   6
           "ring" => R
           "vertices" => {x , x , x , x , x , x }
                           1   2   3   4   5   6

o4 : Graph
 
i5 : allOddHoles H

o5 = {}

o5 : List

See also

Ways to use allOddHoles :

For the programmer

The object allOddHoles is a method function.