Angle Covers: Algorithms and Complexity

Consider a graph with rotation system, namely, for every vertex, circular ordering of the incident edges. Given such graph, an angle cover maps vertex to pair consecutive edges in -- that each edge participates at least one pair. We show any maximum degree 4 admits cover, give poly-time algorithm deciding if no degree-3 vertices has angle-cover, and prove that, given 5, it is NP-hard decide whether cover. also consider extensions problem where selects fixed number $a>1$ angles or consists more than two application covers 2-blowup planar isomorphic thickness 2.