Recontamination Does Not Help to Search a Graph

This paper is concerned with a game on graphs called graph searching. The object of this game is to clear all edges of a contaminated graph. Clearing is achieved by moving searchers, a kind of token, along the edges of the graph according to clearing rules. Certain search strategies cause edges that have been cleared to become contaminated again. Megiddo et al. [9] conjectured that every graph can be searched using a minimum number of searchers without this recontamination occurring, that is, without clearing any edge twice. In this paper, this conjecture is proved. This places the graph-searching problem in NP, completing the proof by Megiddo et al. that the graph-searching problem is NP-complete. Furthermore, by eliminating the need to consider recontamination, this result simplifies the analysis of searcher requirements with respect to other properties of graphs.