SAMS Subject Classification Number: 6

The localization game is played on a graph by two players: a Cop with a team of \(k\) cops, and a Robber. The game is initialised by the Robber choosing a vertex \(r\), unknown to the Cop. Thereafter, the game proceeds turn based. At the start of each turn, the Cop probes k vertices and in return receives a distance vector. If the Cop can determine the exact location of r from the vector, the Robber is located and the Cop wins. Otherwise, the Robber is allowed to either stay at \(r\), or move to a vertex in the neighbourhood of \(r\). Hereafter the Cop again probes \(k\) vertices. The game continues in this fashion, where the Cop wins if the Robber can be located in a finite number of turns. The localization number \(\zeta(G)\), is defined as the least positive integer \(k\) for which the Cop has a winning strategy for the graph \(G\), irrespective of the moves of the Robber.

The localization number of outerplanar graphs have been showed to be either 1 or 2. During this talk we will look at which classes of outerplanar graphs have localization number equal to two.