In the degree-preserving spanning tree problem, we seek a spanning tree with a maximum number of vertices whose incident edges all belong to the spanning tree. It has been recently introduced by Broersma et al. due to a nice application in network flow control. In view of this application, planar bounded-degree graphs deserve particular interest. We prove NP-completeness for bipartite planar degree-5 graphs and for planar degree-3 graphs. This strengthens a result from the mentioned paper. Furthermore, we establish a close relationship to some other well-known graph problems.
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