The path relation for directed planar graphs in rectangles, and its relation to the free diad

In this paper, we define the path relation of a directed graph to be the relation which relates two vertices if there is a path from the first to the second. We study the restriction of this relation to paths from sources to sinks, and consider the question of when two finite graphs embedded in a rectangle give the same relation. We find a set of local changes to these graphs which can be used to get between any two graphs for which this relation is the same. Furthermore, we classify the relations which can arise as this relation for a finite directed graph embedded in a rectangle as the triconvex relations between finite ordinals (defined in this paper). ork originated from some of the author’s work on category theory. It turns out that the category of finite ordinals and relations that can be the path relation of a directed graph embedded in a rectangle, is relevant to the study of diads—introduced by the author as a common generalisation of monads and comonads (note that the terms diad and dyad have been used to mean different things by other authors). More specifically, the referee of one of the author’s papers suggested that it would be useful to identify the category which plays the role for diads that the category of finite ordinals and order-preserving functions plays for monads. It turns out that the category of finite ordinals and relations that can be path relations of graphs embedded in a rectangle, is exactly the category that plays this role.