# Connected incomparability graph

Let $$X$$ be a finite set equipped with a partial order. (Not a preorder: $$a < b$$ implies $$b \not< a$$.) The corresponding incomparability graph has vertex set $$X$$ with an edge between two points iff they are incomparable.

I am interested in posets for which the incomparability graph is connected and are maximal for this property. That is, making any pair of incomparable elements comparable disconnects the incomparability graph.

Is there any hope of classifying such partial orders? For example, the set $$\{a_1, \ldots, a_{n-1},b\}$$ with $$a_i < a_j$$ when $$i < j$$ and $$b$$ incomparable to each $$a_i$$ is maximal in the desired sense. (If we make $$b$$ comparable to some $$a_i$$ then this $$a_i$$ will be comparable to everything and hence an isolated point.) But there are other examples that don't look like this.

Minimal connected graph is a tree, and a co-comparability graph does not contain a subdivided $$K_{1,3}$$ as an induced subgraph; it is denoted by $$T_2$$ on http://www.graphclasses.org/classes/gc_147.html.
Trees with no induced $$T_2$$ are caterpillars (https://en.wikipedia.org/wiki/Caterpillar_tree), and it is easy to see that every caterpillar can be represented as an intersection graph of $$x$$-monotone curves between two vertical lines; in fact, it is a permutation graph (and therefore co-comparability graph).