On the topological testability conjecture for analog fault diagnosis problems

This paper gives a proof of a sharpened version of the conjecture in [1]. Letbf{C}^{r}ber-dimensional complex vector space and letg 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 thek-node fault testability condition [1] is satisfied on an ample subset of values ofg, if, and only if, for any setXof inaccessible nodes, there are at leastk + 1nodes inX^{C}(complement ofX) each of which is connected withXvia 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].