Parameterized complexity of two-interval pattern problem

A 2-interval is the union of two disjoint intervals on real line. Two 2-intervals D1 and D2 are if their intersection empty (i.e., no interval intersects any D2). There can be three different relations between 2-intervals; namely, preceding (<), nested (⊏) crossing (≬). called R-comparable for some R∈{<,⊏,≬}, either D1RD2 or D2RD1. set D R-comparable, R⊆{<,⊏,≬} R≠∅, every pair in R∈R. Given a R⊆{<,⊏,≬}, objective pattern problem to find largest subset that R-comparable. The known W[1]-hard when |R|=3 NP-hard |R|=2 (except R={<,⊏}, which solvable quadratic time). In this paper, we fully settle parameterized complexity by showing it both R={⊏,≬} R={<,≬} (when size an optimal solution). This answers open question posed Vialette ((2008) [22]).