graphicIdeal -- creates the toric ideal of the non-incidence graph of a polytope

Synopsis

• Usage:

  T = graphicIdeal V

  T = graphicIdeal P

  T = graphicIdeal S

  T = graphicIdeal M

• Inputs:
  • V, a list, a list of vertices or of facet sets
  • P, a convex polyhedron, a polytope
  • S, a matrix, a (symbolic) slack matrix of a polytope
  • M, a matroid, a matroid
• Optional inputs:
  • CoefficientRing => ..., default value QQ, specifies the coefficient ring of the underlying ring of the ideal
  • Object => ..., default value "polytope", specify combinatorial object
  • Saturate => ..., default value "all", specifies saturation strategy to be used
  • Strategy => ..., default value Eliminate, specifies saturation strategy to be used
  • Vars => ..., default value {}, specifies the variables to use to create the underlying ring of the ideal
• Outputs:
  • T, an ideal, the toric ideal of the non-incidence graph

Description

i1 : P = convexHull(matrix{{0, 0, 1, 1}, {0, 1, 0, 1}});

i2 : T = graphicIdeal P

Order of vertices is 
{{0, 0}, {1, 0}, {0, 1}, {1, 1}}
Graph computed from symbolic adjacency matrix: | 0   y_1 0   y_2 |
                                               | y_3 0   0   y_4 |
                                               | 0   y_5 y_6 0   |
                                               | y_7 0   y_8 0   |

o2 = ideal(y y y y  - y y y y )
            0 3 5 6    1 2 4 7

o2 : Ideal of QQ[y ..y ]
                  0   7

Caveat

See also

• graphFromSlackMatrix -- creates the vertex-edge incidence matrix for the bipartite non-incidence graph with adjacency matrix the given slack matrix
• cycleIdeal -- constructs the cycle ideal of a realization
• symbolicSlackMatrix -- computes the symbolic slack matrix