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Ad hoc, actuator and sensor wireless networks normally have critical connectivity properties before becoming fault intolerant. Existing algorithms for testing k-connectivity are centralized. In this article, we introduce localized algorithms for testing A-connectivity. In localized protocols, each node makes its own decision based on the information available in its local neighborhood. In the first proposed local neighbor detection (LND) algorithm, each node verifies whether or not itself and each of its p-hop neighbors have at least k neighbors. In the second local critical node detection (LCND) protocol, it also tests if the subgraph of its p-hop neighbours of a given node is k-connected. The third local subgraph connectivity detection (LSCD) protocol is based on communications between neighboring nodes to exchange the local decisions starting from k=l. All nodes declare themselves locally 1-connected. For k=2,3,..., iteratively, local decisions are propagated to p-hop neighbors. If node A is (k-1,)-connected, all its p-hop neighbors are (k-1)-connected, and the graph consisting of p-hop neighbors of A (excluding A) is (k-1,)-connected, then node A declares its neighborhood as k-connected. The experiments are carried with two ways of uniform generation of connected unit disk graphs. They show low percentage of false 'alarms', ability to locate critical areas in k-disconnected networks, and increased accuracy with increased local knowledge.