Definition
As defined, e.g., by Goldberg & Karzanov (1996), a skew-symmetric graph G is a directed graph, together with a function σ mapping vertices of G to other vertices of G, satisfying the following properties:
- For every vertex v, σ(v) ≠ v,
- For every vertex v, σ(σ(v)) = v,
- For every edge (u,v), (σ(v),σ(u)) must also be an edge.
One may use the third property to extend σ to an orientation-reversing function on the edges of G.
The transpose graph of G is the graph formed by reversing every edge of G, and σ defines a graph isomorphism from G to its transpose. However, in a skew-symmetric graph, it is additionally required that the isomorphism pair each vertex with a different vertex, rather than allowing a vertex to be mapped to itself by the isomorphism or to group more than two vertices in a cycle of isomorphism.
A path or cycle in a skew-symmetric graph is said to be regular if, for each vertex v of the path or cycle, the corresponding vertex σ(v) is not part of the path or cycle.
Read more about this topic: Skew-symmetric Graph
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