Topic: st-connectivity, st-cut ideals and path ideals of a graph
My Lemma: None of the st-cut-monomials vanish iff there is at least a st-path that does not vanish.
Example
ST-cuts: {{1,3,5,6},{1,3,7},{1,4,5,6},{1,4,7},{2,3,5,6},{2,3,7},{2,4,5,6},{2,4,7}}
ST-paths: {{1,2},{3,4},{5,7},{6,7}}
ST-cut-monomials: $$\{\{x_1x_3x_5x_6\},\{x_1x_3x_7\},\{x_1x_4x_5x_6\},\{x_1x_4x_7\},\{x_2x_3x_5x_6\},\{x_2x_3x_7\},\{x_2x_4x_5x_6\},\{x_2x_4x_7\}\}$$
ST-path-monomials: $$\{\{x_1x_2\},\{x_3x_4\},\{x_5x_7\},\{x_6x_7\}\}$$
Ideals by the cut monomials and by the path monomials, respectively,
$\begin{align} \langle C_{all}\rangle = \langle \text{cut monomials not vanishing}\rangle &= \tiny{\langle x_1x_3x_5x_6-1,x_1x_3x_7-1,x_1x_4x_5x_6-1,x_1x_4x_7-1,x_2x_3x_5x_6-1,x_2x_3x_7-1,x_2x_4x_5x_6-1,x_2x_4x_7-1\rangle } \\ \langle P_{all}\rangle = \langle \text{path monomials not vanishing}\rangle &= \langle x_1x_2 -1,x_3x_4 -1,x_5x_7-1,x_6x_7 -1 \rangle \end{align}$
and $\langle C_{all}\rangle =\langle P_{all}\rangle$ is equivalent to
None of cut monomials vanish iff none of path monomials vanish.so$\langle C_{all}\rangle = \langle x_1x_2 -1\rangle$
$\langle C_{all}\rangle = \langle x_3x_4 -1\rangle$
$\langle C_{all}\rangle = \langle x_5x_7-1\rangle$
$\langle C_{all}\rangle = \langle x_6x_7 -1\rangle$
where any of four expressions is sufficient that Out-In are connected.
How to express the duality between cuts and paths in terms of st-cut-ideals and st-path-ideals?