Planar subgraphs without low-degree nodes
We study the following problem: given a geometric graph G and an integer k, determine if G has a planar spanning subgraph (with the original embedding and straight-line edges) such that all nodes have degree at least k. If G is a unit disk graph, the problem is trivial to solve for k = 1. We show that even the slightest deviation from the trivial case (e.g., quasi unit disk graphs or k = 2) leads to NP-hard problems.
Kranakis, E, Morales Ponce, O. (Oscar), & Suomela, J. (Jukka). (2011). Planar subgraphs without low-degree nodes. doi:10.1007/978-3-642-22300-6_49