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.
Dichotomies for maximum matching cut:H-freeness, bounded diameter, bounded radius.Theoretical Com- puter Science, 1017:114795
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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.