How to find implication classes in a graph?

Question

I understood the general idea of comparability graphs and transitive orientation but just can't wrap my head around the implication classes.
From Advanced Topics in Graph Algorithms - Ron Shamir.

How exactly these implication classes are constructed ?

Answer 1

Checking the definition, we can build the following table for which edges directly force which other edges in Figure 6.5:

\begin{array}{c|ccccccc} \Gamma & ab & ac & ad & ae & bc & bd & be & cd \\ \hline ab & & \\ ac & & & & \times \\ ad & & & & \times \\ ae & & \times & \times \\ bc & & & & & & & \times \\ bd & & & & & & & \times \\ be & & & & & \times & \times \\ cd \\ \end{array} Taking the reflexive-transitive closure of this relation gives us the following:

\begin{array}{c|ccccccc} \Gamma^* & ab & ac & ad & ae & bc & bd & be & cd \\ \hline ab &\times & \\ ac & & \times & \times &\times \\ ad & & \times & \times &\times \\ ae & & \times & \times &\times \\ bc & & & & & \times & \times & \times \\ bd & & & & & \times & \times & \times \\ be & & & & & \times & \times & \times \\ cd & & & & & & & & \times \\ \end{array}

The equivalence classes of $\Gamma^*$ are therefore $\{ab\}$, $\{ac, ad, ae\}$, $\{bc,bd,be\}$, and $\{cd\}$.