In this paper we consider a fundamental problem in the area of viral marketing, called TARGET SET SELECTION problem. We study the problem when the underlying graph is a block-cactus graph, a chordal graph or a Hamming graph. We show that if G is a block-cactus graph, then the TARGET SET SELECTION problem can be solved in linear time, which generalizes Chen's result [2] for trees, and the time complexity is much better than the algorithm in [1] (for bounded treewidth graphs) when restricted to block-cactus graphs. We show that if the underlying graph G is a chordal graph with thresholds θ(v) ≤ 2 for each vertex v in G, then the problem can be solved in linear time. For a Hamming graph G having thresholds θ(v) = 2 for each vertex v of G, we precisely determine an optimal target set S for (G, θ). These results partially answer an open problem raised by Dreyer and Roberts [3].

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