Fast Searching on k-Combinable Graphs

Finding an optimal fast search strategy for graphs is challenging, sometimes even when graphs have very small treewidth, like cacti, cartesian product of a tree and an edge, etc. However, it may be easier to find an optimal fast search strategy for some critical subgraphs of the given graph. Although fast searching is not subgraph-closed, this observation still motivates us to establish relationships between optimal fast search strategies for a graph and its subgraphs. In this paper, we introduce the notion of k-combinable graphs and propose a new method for computing their fast search number. Assisted by the new method, we investigate the fast search number of cacti graphs and the cartesian product of a tree and an edge. Algorithms for producing fast search strategies for the above graphs, along with rigorous proofs, are given in this paper.