Skew-symmetric Graph - Polar/switch Graphs, Double Covering Graphs, and Bidirected Graphs

Polar/switch Graphs, Double Covering Graphs, and Bidirected Graphs

A skew-symmetric graph may equivalently be defined as the double covering graph of a polar graph (introduced by Zelinka (1974), Zelinka (1976), called a switch graph by Cook (2003)), which is an undirected graph in which the edges incident to each vertex are partitioned into two subsets. Each vertex of the polar graph corresponds to two vertices of the skew-symmetric graph, and each edge of the polar graph corresponds to two edges of the skew-symmetric graph. This equivalence is the one used by Goldberg & Karzanov (1996) to model problems of matching in terms of skew-symmetric graphs; in that application, the two subsets of edges at each vertex are the unmatched edges and the matched edges. Zelinka (following F. Zitek) and Cook visualize the vertices of a polar graph as points where multiple tracks of a train track come together: if a train enters a switch via a track that comes in from one direction, it must exit via a track in the other direction. The problem of finding non-self-intersecting smooth curves between given points in a train track comes up in testing whether certain kinds of graph drawings are valid (Hui, Schaefer & Štefankovič 2004) and may be modeled as the search for a regular path in a skew-symmetric graph.

A closely related concept is the bidirected graph of Edmonds & Johnson (1970) ("polarized graph" in the terminology of Zelinka (1974), Zelinka (1976)), a graph in which each of the two ends of each edge may be either a head or a tail, independently of the other end. A bidirected graph may be interpreted as a polar graph by letting the partition of edges at each vertex be determined by the partition of endpoints at that vertex into heads and tails; however, swapping the roles of heads and tails at a single vertex ("switching" the vertex, in the terminology of Zaslavsky (1991)) produces a different bidirected graph but the same polar graph.

For the correspondence between bidirected graphs and skew-symmetric graphs (i.e., their double covering graphs) see Zaslavsky (1991), Section 5, or Babenko (2006).

To form the double covering graph (i.e., the corresponding skew-symmetric graph) from a polar graph G, create for each vertex v of G two vertices v₀ and v₁, and let σ(v_i) = v_{1 − i}. For each edge e = (u,v) of G, create two directed edges in the covering graph, one oriented from u to v and one oriented from v to u. If e is in the first subset of edges at v, these two edges are from u₀ into v₀ and from v₁ into u₁, while if e is in the second subset, the edges are from u₀ into v₁ and from v₀ into u₁. In the other direction, given a skew-symmetric graph G, one may form a polar graph that has one vertex for every corresponding pair of vertices in G and one undirected edge for every corresponding pair of edges in G. The undirected edges at each vertex of the polar graph may be partitioned into two subsets according to which vertex of the polar graph they go out of and come in to.

A regular path or cycle of a skew-symmetric graph corresponds to a path or cycle in the polar graph that uses at most one edge from each subset of edges at each of its vertices.