Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Some NP-complete edge packing and partitioning problems in planar graphs   Graph packing and partitioning problems have been studied in many contexts,
including from the algorithmic complexity perspective. Consider the packing
problem of determining whether a graph contains a spanning tree and a cycle
that do not share edges. Bern\'ath and Kir\'aly proved that this decision
problem is NP-complete and asked if the same result holds when restricting to
planar graphs. Similarly, they showed that the packing problem with a spanning
tree and a path between two distinguished vertices is NP-complete. They also
established the NP-completeness of the partitioning problem of determining
whether the edge set of a graph can be partitioned into a spanning tree and a
(not-necessarily spanning) tree. We prove that all three problems remain
NP-complete even when restricted to planar graphs.

23
The categories of this paper are: solv-int
22
The categories of this paper are: chao-dyn
3
The categories of this paper are: physics
27
The categories of this paper are: comp-gas
12
The categories of this paper are: nucl-th
0
The categories of this paper are: hep-ph
32
The categories of this paper are: atom-ph
13
The categories of this paper are: quant-ph
36
The categories of this paper are: cmp-lg
14
The categories of this paper are: math-ph
37
The categories of this paper are: supr-con
11
The categories of this paper are: hep-lat
34
The categories of this paper are: chem-ph
28
The categories of this paper are: alg-geom
16
The categories of this paper are: stat
10
The categories of this paper are: q-bio