A theory of structural vulnerability is developed and presented in two linked papers, initially for two dimensional frame structures. The purpose of the theory is to enable the identification of the most vulnerable parts of a structural system so that they may be suitably protected and monitored. In Part 1 a graph model is used to analyse various load paths and loops within a structure. The most important new concept presented is that of a structural ring. A structural system is represented at various hierarchical levels of definition in terms of clusters of structural rings. A measure of the quality of well-formedness of structural rings and clusters is developed which in Part 2 will be used as a quantitative estimate of structural vulnerability.