Every triangle-free planar graph has a planar upward drawing (Q1311415)

A planar upward drawing means a Hasse diagram of a partially ordered set. It is said that a planar graph \(G\) has a planar upward drawing, if it is possible to assign directions to its edges in such a way that a directed graph of a partially ordered set (shortly poset) is obtained and if this graph can be drawn in the plane as the Hasse diagram of this poset. A directed graph of a poset is the graph whose vertex set is that poset and in which an edge from \(x\) to \(y\) exists if and only if \(y\) covers \(x\) in the poset. To draw such a graph as the Hasse diagram means to draw it in such a way that \(y\) is in a higher position than \(x\) whenever \(x<y\). The main result of this paper states that every triangle-free planar graph has a planar upward drawing.