Fast searching on cactus graphs

The problem of finding the fast search number of a graph is NP-complete. It is challenging even when the graph has very small treewidth. However, it can be much easier to find an optimal fast search strategy for smaller subgraphs with special properties. This observation motivates us to establish relationships between optimal fast search strategies for a graph and its subgraphs although fast searching does not have the subgraph-closed property. In this paper, we introduce the notion of k -combinable graphs and study their properties. We propose a new method for computing the fast search number of k -combinable graphs. As an application of this method, we examine the fast searching for cactus graphs. We investigate the properties of optimal fast search strategies and give a linear time algorithm for computing the fast search number of cactus graphs.