Conflict-free cuts exist in all planar 4-regular graphs with 1-regular conflicts except the octahedron, but the decision problem is NP-complete for planar and 3-degenerate graphs of maximum degree 5.
On conflict-free spanning tree: Mapping tractable and hard instances through the lenses of graph classes.Theoretical Computer Science, 1031:115081
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Conflict-Free Cuts in Planar and 3-Degenerate Graphs with 1-Regular Conflicts
Conflict-free cuts exist in all planar 4-regular graphs with 1-regular conflicts except the octahedron, but the decision problem is NP-complete for planar and 3-degenerate graphs of maximum degree 5.