Some Results on the Target Set Selection Problem

In this paper we consider a fundamental problem in the area of viral marketing, called T{\scriptsize ARGET} S{\scriptsize ET} S{\scriptsize ELECTION} 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 T{\scriptsize ARGET} S{\scriptsize ET} S{\scriptsize ELECTION} problem can be solved in linear time, which generalizes Chen's result \cite{chen2009} for trees, and the time complexity is much better than the algorithm in \cite{treewidth} (for bounded treewidth graphs) when restricted to block-cactus graphs. We show that if the underlying graph $G$ is a chordal graph with thresholds $\theta(v)\leq 2$ for each vertex $v$ in $G$, then the problem can be solved in linear time. For a Hamming graph $G$ having thresholds $\theta(v)=2$ for each vertex $v$ of $G$, we precisely determine an optimal target set $S$ for $(G,\theta)$. These results partially answer an open problem raised by Dreyer and Roberts \cite{Dreyer2009}.