The planar minimum linear arrangement problem is different from the minimum linear arrangement problem

In various research papers, such as [2], one will find the claim that the minLA is optimally solvable on outerplanar graphs, with a reference to [1]. However, the problem solved in that publication, which we refer to as the planar minLA, is different from the minLA, as we show in this article. In constrast to the minimum linear arrangement problem (minLA), the planar minimum linear arrangement problem (planar minLA) poses an additional restriction on the arrangements: It must be possible to draw all edges of the input graph G such that they “run above” the nodes and do not intersect. More formally: Definition 1 (Crossing edges). Let G = (V,E) be a graph and let π be a linear arrangement of G. Two distinct edges {u, v}, {x, y} ∈ E cross if: π(u) < π(x) < π(v) < π(y). Definition 2 (Minimum planar arrangement). A minimum planar arrangement of an input graph G = (V,E) is a mapping π : V → {1, . . . , n} such that no two edges of G cross in π. We prove that optimal solutions of the planar minLA are different from optimal solutions of the minLA by presenting a counterexample. That is, we give an example graph whose corresponding minimum linear arrangement yields a smaller cost than all possible minimum planar arrangements. The input graph G = (V,E) we use is given by Figure 1.