Vertices of Localized Imbalance in a Biased Graph

A biased graph consists of a graph T and a subclass B of the polygons of T, such that no theta subgraph of T contains exactly two members of B. A subgraph is balanced when all its polygons belong to B. A vertex is a balancing vertex if deleting it leaves a balanced graph. We give a construction for unbalanced biased graphs having a balancing vertex and we show that an unbalanced biased graph having more than one balancing vertex is an unbalanced series or parallel connection of balanced graphs. Introduction. A polygon in a graph T is the edge set of a simple closed path. A subclass of polygons of T is a linear subclass if no theta subgraph of T contains exactly two polygons in the subclass. A biased graph consists of a graph T and a linear subclass S of polygons of T [2]. A subgraph or edge set is balanced if every polygon in it belongs to B. Biased graphs are a generalization of ordinary graphs, in the sense that for most purposes the latter can be treated as biased graphs that are balanced. For instance, a biased graph (r, B) has a "bias matroid" G(T, B) and a "lift matroid" L(T,B) (see [3]), which when (T,B) is balanced equal the usual polygon matroid of T. We say that the imbalance of a biased graph (T,B) is localized at a vertex v if the vertex-deleted subgraph T \ v is balanced. Briefly, we call v a balancing vertex. We are interested in biased graphs that have balancing vertices because in several senses they are among the least unbalanced. For one thing, by definition they are the unbalanced graphs in which balance results by deleting the fewest vertices. They are also the simplest kind of unbalanced graph having no two vertex-disjoint unbalanced polygons. Finally, they are among those biased graphs whose bias and lift matroids are graphic matroids. In this article we see that the set V* of balancing vertices of (T,B) is usually empty and we describe the exceptions (see the Theorem and Corollary 2). Definitions and a lemma. We let T denote a graph (V, E) with vertex set V and edge set E. All graphs are finite. Loops and multiple edges are allowed. By mKi we mean m links (edges with two distinct endpoints) in parallel on two vertices. By the polygon graph C¡ we mean the graph of a simple, closed path of length /; in particular, C2 = 2K2. Received by the editors November 20, 1985 and, in revised form, June 14, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 05C99; Secondary 05C25.