As for the categories of subgraphs of a given graph $X$ with their inclusions, we have a co-Heyting algebra (and Heyting algebra).
In the co-Heyting algebra, we define $\sim$$A$ as the pseudo-complement of a subgraph A as follows: it is the smallest subgraph whose union with A is equal to the whole given graph, in symbol $\sim$$A\vee$$A=X$.
I'm wondering whether the double pseudo-complement $\sim$$(\sim$$A)=A$ holds or not. I have a difficulty in either of proving $\sim$$(\sim$$A)=A$ or providing a counter example. Based on a few experiments in my notepad, it seems that $\sim$$(\sim$$A)=A$ may hold.
My question is, does $\sim$$(\sim$$A)=A$ hold? If so, how can I prove it? If not, can I have a counter example?
This is not true; the double pseudo-complement will erase any “isolated” node from the original subgraph. Consider (undirected) 3-cycle (or complete graph on $3$ nodes) $C_3$ with set of vertices $V=\{0,1,2\}$, and take the subgraph $A:=(V,\{\{0,1\}\})$. Then ${\sim} A=(V,\{\{0,2\},\{1,2\}\})$ and ${\sim}{\sim} A=(\{0,1\},\{\{0,1\}\})$.