Abstract:
We propose a new model for studying graph related problems that we call the \emph{orientation model}. In this model, an undirected graph $G$ is fixed, and the input is any possible edge orientation of $G$. A property is now a property of the directed graph that is obtained by a given orientation. The distance between two orientations is the number of edges that have to be redirected in order to move from one digraph to the other. This model allows studying digraph properties such as not containing a forbidden (induced) subgraph, being strongly connected etc., for every underlying graph including sparse graphs. As it turns out, this model generalizes the standard, adjacency matrix model. That is, we show that for every graph property $\mathcal{P}$ of dense graphs there is a property of orientations that is testable if and only if $\mathcal{P}$ is. This model is also handy in some practical situations of networks, in which the underlying network is fixed while the direction of (weighted) links may vary. We show that several orientations properties are testable in this model (for every underlying graph), while some are not.