Some theoretical results on sensor placement for diagnosis based on fault isolability specifications

The objective of this section is to introduce notation and theoretical tools regarding structural models and bipartite graphs that will be used in proofs of results in coming section. The structural representation of a set of equations M with unknown variables X is a bipartite graph with variables and equations as node sets. There is an edge in the graph between a node representing an equation e ∈ M and node representing an unknown variable x ∈ X if the variable x is contained in e. For notational convenience, we will denote the node representing an equation e or a variable x simply by the equation name e and the variable name x respectively. A bipartite graph can be described by a biadjacency matrix where the rows and columns correspond to the node sets and the position (i, j) is one if there is an edge between node i and j, and a zero otherwise.