On the topological testability conjecture for analog fault diagnosis problems

This paper gives a proof of a sharpened version of the conjecture in [1]. Let
\bf{C}^{r}
be
r
-dimensional complex vector space and let
g \in {\bf C}^{r}
be the vector of branch admittances of an analog network. A subset of
{\bf C}^{r}
is said to be ample if (i) its complement has Lebesgue measure zero, (ii) it is open, and (iii) it is dense. The sharpened version of the conjecture claims that the
k
-node fault testability condition [1] is satisfied on an ample subset of values of
g
, if, and only if, for any set
X
of inaccessible nodes, there are at least
k + 1
nodes in
X^{C}
(complement of
X
) each of which is connected with
X
via a branch. This is extremely powerful because the result depends only on the topology of a network and the condition can be checked by inspection. The proof justifies the fault location method developed in [1].