Biased Graph - Examples

Examples

• If every circle belongs to B, and there are no half-edges, Ω is balanced. A balanced biased graph is (for most purposes) essentially the same as an ordinary graph.

• If B is empty, Ω is called contrabalanced. Contrabalanced biased graphs are related to bicircular matroids.

• If B consists of the circles of even length, Ω is called antibalanced and is the biased graph obtained from an all-negative signed graph.

• The linear class B is additive, that is, closed under repeated symmetric difference (when the result is a circle), if and only if B is the class of positive circles of a signed graph.

• Ω may have underlying graph that is a cycle of length n ≥ 3 with all edges doubled. Call this a biased 2C_n . Such biased graphs in which no digon (circle of length 2) is balanced lead to spikes and swirls (see Matroids, below).

• Some kinds of biased graph are obtained from gain graphs or are generalizations of special kinds of gain graph. The latter include biased expansion graphs, which generalize group expansion graphs.