Node Failure Localization: Theorem Proof

For Algorithm 3, the input network is not necessarily a connected graph. Lemma II.1, however, only considers the case that the input network satisfying the three conditions (in Lemma II.1) is connected. Then it suffices to show that Algorithm 3 places the minimum number of monitors to ensure that any two non-monitors in G′ (satisfying the three conditions in Lemma II.1) are distinguishable. Such connected Algorithm 3: Monitors-in-Polygon-less-Network(G, S) input : Network topology G, node set S output: Sub-set of nodes in G as monitors 1 if |L| = 0 then 2 return; 3 end 4 foreach connected component Gi in G do 5 if Gi contains only one biconnected component then 6 randomly choose a node in Gi as a monitor; 7 else 8 in Gi, label one biconnected component with 0 or 1 cut-vertex as B1, one neighboring biconnected component of B1 as B2 (if any), and one neighboring biconnected component of B2 other than B1 as B3 (if any); 9 if B2 is a bond then 10 choose the common node between B1 and B2 as a monitor;