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.