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Non-adaptive group testing involves grouping arbitrary subsets of $n$ items into different pools. Each pool is then tested and defective items are identified. A fundamental question involves minimizing the number of pools required to identify at most $d$ defective items. Motivated by applications in network tomography, sensor networks and infection propagation, a variation of group testing problems on graphs is formulated. Unlike conventional group testing problems, each group here must conform to the constraints imposed by a graph. For instance, items can be associated with vertices and each pool is any set of nodes that must be path connected. In this paper, a test is associated with a random walk. In this context, conventional group testing corresponds to the special case of a complete graph on $n$ vertices. For interesting classes of graphs a rather surprising result is obtained, namely, that the number of tests required to identify $d$ defective items is substantially similar to what required in conventional group testing problems, where no such constraints on pooling is imposed. Specifically, if $T(n)$ corresponds to the mixing time of the graph $G$, it is shown that with $m=O(d^2T^2(n)\log(n/d))$ non-adaptive tests, one can identify the defective items. Consequently, for the Erd\H{o}s-R\'enyi random graph $G(n,p)$, as well as expander graphs with constant spectral gap, it follows that $m=O(d^2\log^3n)$ non-adaptive tests are sufficient to identify $d$ defective items. Next, a specific scenario is considered that arises in network tomography, for which it is shown that $m=O(d^3\log^3n)$ non-adaptive tests are sufficient to identify $d$ defective items. Noisy counterparts of the graph constrained group testing problem are considered, for which parallel results are developed.