Biased graph

In mathematics, a biased graph is a graph with a list of distinguished circles (edge sets of simple cycles), such that if two circles in the list are contained in a theta graph, then the third circle of the theta graph is also in the list. A biased graph is a generalization of the combinatorial essentials of a gain graph and in particular of a signed graph.

Formally, a biased graph Ω is a pair (G, B) where B is a linear class of circles; this by definition is a class of circles that satisfies the theta-graph property mentioned above.

A subgraph or edge set whose circles are all in B (and which contains no half-edges) is called balanced. For instance, a circle belonging to B is balanced and one that does not belong to B is unbalanced.

Biased graphs are interesting mostly because of their matroids, but also because of their connection with multiary quasigroups. See below.

A biased graph may have half-edges (one endpoint) and loose edges (no endpoints). The edges with two endpoints are of two kinds: a link has two distinct endpoints, while a loop has two coinciding endpoints.

Linear classes of circles are a special case of linear subclasses of circuits in a matroid.

A minor of a biased graph Ω = (G, B) is the result of any sequence of taking subgraphs and contracting edge sets. For biased graphs, as for graphs, it suffices to take a subgraph (which may be the whole graph) and then contract an edge set (which may be the empty set).

A subgraph of Ω consists of a subgraph H of the underlying graph G, with balanced circle class consisting of those balanced circles that are in H. The deletion of an edge set S, written Ω − S, is the subgraph with all vertices and all edges except those of S.

...
Wikipedia