Biased graphs. I: Bias, balance, and gains
A biased graph is a graph with a class of its circles (simple closed paths), called balanced, having the property that, whenever the union of two balanced circles is a theta graph, the third circle in the union is also balanced. A gain graph is a graph in which each edge has a gain, i.e. a value from a group so that reversing the direction inverts the gain, a circle is balanced if its edge gain product is the identity of the group. So, a gain graph defines a biased graph. (The gain graphs form a proper subclass of biased graphs.) This paper is the first part of the series in which the general structural and enumerative theory of biased-graphic matroids is developed. In this part, the fundamentals are presented of balance and minors (subgraphs and contractions) for biased and gain graphs, the parts that are presupposed by later parts. There is also the catalogue of the most interesting general types of biased and gain graphs followed by examples.
scientific article; zbMATH DE number 4174986
scientific article