Short proofs on the structure of general partition, equistable and triangle graphs

While presenting a combinatorial point of view to the class of equistable graphs, Miklavič and Milanič pointed out the inclusions among the classes of equistable, general partition and triangle graphs. Orlin conjectured that the first two classes were equivalent, but in 2014, Milanič and Trotignon showed that the three classes are all distinct, leading to the following hierarchy: general partition graphs ⊂ equistable graphs ⊂ triangle graphs. In this paper, we solve an open problem from Anbeek et al. (1997) by showing that all the above classes are equivalent when restricted to planar graphs. Additionally, we present short proofs on some structural properties of the general partition, equistable and triangle classes. In particular, we show that the general partition and triangle classes are both closed under the operations of substitution, induction and contraction of modules which allow us to recover several results in the literature.