Searching graphs
Edge searching is a pursuit-evasion game played on graphs. Scientists from diverse fields have been devoted to studying this interdisciplinary field over the last two decades. One of the main challenges is to characterise graphs in which the edge number is equivalent to the searchers' number. The EU-funded project 'Forbidden Minor characterizations for 4-searchable graphs' (FORMI 4-SEG) has addressed this high-priority topic in mathematics. Equivalently, the project aims to clean a tunnel network filled with a noxious gas using as few cleaners as possible. In particular, the project aims to characterise graph minors where k number is greater than three. Such graphs — also called obstruction sets — can be produced by edge contraction or vertex deletion of a given graph. FORMI 4-SEG assumes that the graph is initially entirely contaminated. The cleaning task is performed by a searchers' team that are allowed to make three types of moves: a searcher is placed on a vertex, a searcher is removed from a vertex and a searcher slides from a vertex to one of its neighbours. Scientists constructed forbidden minors for biconnected four-edge searchable series-parallel graphs. This method was extended for k-edge searchable series-parallel graphs. Another important result was employing a general method that is called k-sums for constructing forbidden minors for a minor closed family. The main expected result is to construct a four-edge searchable graph that does not contain any obstruction sets. Project results opened the topic to much more research and interpretation in the mathematics field with applications including structure identification.
