Permutation reconstruction from MinMaxMinMax-Betweenness constraints

In this paper, we investigate the reconstruction of permutations on {1,2,…,n}{1,2,…,n} from betweenness constraints involving the minimum and the maximum element located between tt and t+1t+1, for all t=1,2,…,n−1t=1,2,…,n−1. We propose two variants of the problem (directed and undirected), and focus first on the directed version, for which we draw up general features and design a polynomial algorithm in a particular case. Then, we investigate necessary and sufficient conditions for the uniqueness of the reconstruction in both directed and undirected versions, using a parameter kk whose variation controls the stringency of the betweenness constraints. We finally point out open problems.