In Memoriam Walter Kern

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