Shortest reconfiguration of sliding tokens on subclasses of interval graphs

Abstract

Suppose that two independent sets Ib and Ir of a graph such that |Ib|=|Ir| are given, and a token is placed on each vertex in Ib. The sliding token problem is to determine whether there exists a sequence of independent sets which transforms Ib into Ir so that each independent set in the sequence results from the previous one by sliding exactly one token along an edge in the graph. The sliding token problem is one of the reconfiguration problems that attract the attention from the viewpoint of theoretical computer science. Recently, the problems that aim at finding a shortest reconfiguration sequence are investigated. In general, even if it is polynomial time solvable to decide whether two instances are reconfigurable into each other, it can be NP-hard to find a shortest sequence between them. In this paper, we show that the problem for finding a shortest sequence between two independent sets is polynomial time solvable for some graph classes which are subclasses of the class of interval graphs. As far as the authors know, this is the first polynomial time algorithm for the shortest sliding token problem for a graph class that requires detours.