Polynomially tractable cases in the popular roommates problem

The input of the popular roommates problem consists of a graph G = (V, E) and for each vertex v in V, strict preferences over the neighbors of v. Matching M is more popular than M' if the number of vertices preferring M to M' is larger than the number of vertices preferring M' to M. A matching M is called popular if there is no matching M' that is more popular than M. Faenza et al. (2019) and Gupta et al. (2021) proved that determining the existence of a popular matching in a popular roommates instance is NP-complete. In this paper we identify a class of instances that admit a polynomial-time algorithm for the problem. We also test these theoretical findings on randomly generated instances to determine the existence probability of a popular matching in them.