Pfafian orientation of a graph

Pfafian orientation of a graph --- пфафианова ориентация графа.

Let [math]\displaystyle{ G }[/math] be a graph, and [math]\displaystyle{ H }[/math] be a subgraph of [math]\displaystyle{ G }[/math]. We say that [math]\displaystyle{ H }[/math] is central if [math]\displaystyle{ G \setminus V(H) }[/math] has a perfect matching. Let [math]\displaystyle{ D }[/math] be an orientation of [math]\displaystyle{ G }[/math], and let [math]\displaystyle{ C }[/math] be a circuit of [math]\displaystyle{ G }[/math] of even length. We say that [math]\displaystyle{ C }[/math] is oddly oriented (in [math]\displaystyle{ D }[/math]), if [math]\displaystyle{ C }[/math] contains an odd number of edges that are directed (in [math]\displaystyle{ D }[/math]) in the direction of each orientation of [math]\displaystyle{ C }[/math]. We say that [math]\displaystyle{ D }[/math] is a Pfafian orientation of [math]\displaystyle{ G }[/math], if every central circuit of [math]\displaystyle{ G }[/math] of even length is oddly oriented in [math]\displaystyle{ D }[/math].

It is known that a bipartite graph admits a Pfafian orientation if and only if it does not contain [math]\displaystyle{ K_{3,3} }[/math].