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 so is the third circle of the theta graph. 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.
Read more about Biased Graph: Technical Notes, Examples, Minors, Matroids, Multiary Quasigroups
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