Independence is a unifying concept for linear algebra and graphs. A deep generalization of both through this unification, is contained in the notion of graphoids. Every graph can be seen as an interpretation of a graphoid in a particular ‘coordinate system’, called a 2-complete basis. From this prospect, a graphoid is an essential, coordinate free, geometrical notion for which each associated graph, if it exists, is just a particular view of the same generality. The concept of a graphoid can also be seen as a pair of set systems (dual matroids) whose members are called circuits and cutsets. The set of all circuits (cutsets) together with all their distinct unions we call a circ (cut) space. In this chapter, in the context of circuits and cutsets, we concentrate on two concepts of independence within graphs and graphoids. We first introduce independent collections of circuits and cutsets and then we use this concept to define independent edge subsets, that is, circuit-less and cutset-less subsets. Prom this point on, we take circuits and cutsets as primary notions. This material may be seen as a bridge between traditional graph theory and matroid theory. We also give a brief overview of properties of graphoids and methods in topological analysis of networks.

The graphoidal point of view

The space of all graphs can be divided into disjoint classes such that two graphs belong to the same class if they are 2-isomorphic.

This research monograph is concerned with two dual structures in graphs. These structures, one based on the concept of a circuit and the other on the concept of a cutset are strongly interdependent and constitute a hybrid structure called a graphoid. This approach to graph theory dealing with graphoidal structures we call hybrid graph theory. A large proportion of our material is either new or is interpreted from a fresh viewpoint. Hybrid graph theory has particular relevance to the analysis of (lumped) systems of which we might take electrical networks as the archetype. Electrical network analysis was one of the earliest areas of application of graph theory and it was essentially out of developments in that area that hybrid graph theory evolved. The theory emphasises the duality of the circuit and cutset spaces and is essentially a vertex independent view of graphs. In this view, a circuit or a cutset is a subset of the edges of a graph without reference to the endpoints of the edges. This naturally leads to working in the domain of graphoids which are a generalisation of graphs. In fact, two graphs have the same graphoid if they are 2-isomorphic and this is equivalent to saying that both graphs (within a one-to-one correspondence of edges) have the same set of circuits and cutsets.

Historically, the study of hybrid aspects of graphs owes much to the foundational work of Japanese researchers dating from the late 1960's. Here we omit the names of individual researchers, but they may be readily identified through our bibliographic notes.

First two chapters could be seen as a bridge between traditional graph theory and the graphoidal perspective.

The concept of graph inherently includes two dual structures, one based on circuits and the other based on cutsets. These two structures are strongly interdependent. They constitute the so-called graphoidal structure which is a deep generalization of the concept of graph and comes from matroid theory. The main difference between graph and graphoid is that the latter requires no concept of vertex while the first presumes vertex as a primary notion. After some bridging material, our approach will be entirely vertex-independent. We shall concentrate on the hybrid aspects of graphs that naturally involve both dual structures. Because of this approach, the material of this text is located somewhere between graph theory and the theory of matroids. This enables us to combine the advantages of both an intuitive view of graphs and formal mathematical tools from matroid theory.

Starting with the classical definition of a graph in terms of vertices and edges we define circs and cuts and then circuits and cutsets. In this context, a circuit (respectively, cutset) is a minimal circ (cut) in the sense that no proper subset of it is also a circ (cut). We consider some collective algebraic properties and mutual relationships of circs and cuts. Vertex and edge-separators are introduced and through these we define various kinds of connectivity. The immediate thrust is towards a vertex-independent description of graphs, so that later all theorems and propositions will be vertex-independent.

The last two sections of the chapter are devoted to the notions of multiports and Kirchhoff's laws which are basic concepts in network analysis.

Many properties of pairs of trees of a graph are related to the Hamming distance between them. This is important for several graph-theoretical concepts that have featured in hybrid graph theory. Here the notions of perfect pairs and superperfect pairs of trees have played a part. We define and characterize these notions in this chapter and describe necessary conditions for the unique solvability of affine networks in terms of trees and pair of trees.

The small number of theorems and propositions collected together in the opening paragraphs of Chapter 3 will again be frequently referred to here. Familiarity with the basic concepts of graphs such as circuit and cutset are presumed in this chapter. A maximal circuit-less subset of a graph G is called a tree of G while a maximal cutset-less subset of edges is called a cotree. These terms (circuit, cutset, tree and cotree) will be used here to mean a subset of the edges of a graph. Let F be a subset of E. Then the rank of F, denoted by rank (F), is the cardinality of a maximum circuit-less subset of F and the corank of F, denoted by corank (F), is the cardinality of a maximum cutsetless subset of F. The complement of F is the set difference E\F, denoted by F*. By |F| we denote the number of elements in (that is, the cardinality of) the subset F.

Diameter of a tree

Given a graph G, each its tree t can be classified according to the non-negative integer rank(f).

In this chapter maximal edge subsets that are both circuit-less and cutsetless (which we call basoids) and the related concepts of principal minor and principal partition of a graph are considered. The fact that basoids may have different cardinalities provides a rich structure which is described through several propositions. Transitions from one basoid to another (which provides a basis for augmenting basoids in turn) and the concept of a minor of a graph with respect to a dyad (a maximum cardinality basoid) are also investigated in detail. It is shown that there exists a unique minimal minor with respect to every dyad of a graph G.This edge subset, called the principal minor and its dual called the principal cominor, define a partition of the edge set of Gcalled the principal partition. The hybrid rank of a graph is defined to be the cardinality of a dyad of the graph. This is a natural extension of the definitions of rank and corank of a graph, defined as the cardinalities of a maximum circuit-less subset and a maximum cutset-less subset of the graph, respectively. In the last section of this chapter an application to hybrid topological analysis of networks is considered. An algorithm for finding a maximum cardinality topologically complete set of network variables is also described.

The material of this chapter is general in the sense that it can be easily extended from graphs to matroids. To ensure this generality, the Painting Theorem, the matroidal version of the Orthogonality Theorem as well as the Circuit and the Cutset axioms are widely used to prove propositions.