A Catalog of Minimal Blocks * Arthur

The majority of the definitions used here will be found in (3),1 with the terms "point", "line", and "cycle" replaced by vertex, edge, and circuit. In particular, a block is a connected graph with no cut vertex, and a block is minimal if no spanning subgraph of the block with fewer edges is also a block. We consider the graph with one vertex and no edges (the vertex graph) and the graph with two vertices and a single edge joining them (the link graph) to be minimal blocks. To distinguish between a path p and the graph which contains exactly the edges and vertices of p, we denote the graph by Ipl; for simplicity of terminology, we will refer to both p and Ipl as "paths." If p is a path, F (p) denotes the first vertex of p, and L (p) denotes the last vertex of p. The undirected edge joining vertices p and (J" is denoted by (p, (J") or ((J", p) interchangeably. Let A!, . .. , Ak be k disjoint graphs which are paths, with k ;;,; 2, such that path A i contains mi vertices, for each i E {I, ... , k}, with m, ;;,; m2 ;;,; ... ;;,; mk ;;,; 1. Let a and,B be vertices not in k U V(Ai). For each i E {I, ., k}, let ai be a Hamiltonian path inAi. Let the graph P(ml, m2, .. "' i='