A Complexity Dichotomy for Colourful Components Problems on $k$-caterpillars and Small-Degree Planar Graphs

A connected component of a vertex-coloured graph is said to be colourful if all its vertices have different colours, and a graph is colourful if all its connected components are colourful. Given a vertex-coloured graph, the Colourful Components problem asks whether there exist at most $p$ edges whose removal makes the graph colourful, and the Colourful Partition problem asks whether there exists a partition of the vertex set with at most $p$ parts such that each part induces a colourful component. We study the problems on $k$-caterpillars (caterpillars with hairs of length at most $k$) and explore the boundary between polynomial and NP-complete cases. It is known that the problems are NP-complete on $2$-caterpillars with unbounded maximum degree. We prove that both problems remain NP-complete on binary $4$-caterpillars and on ternary $3$-caterpillars. This answers an open question regarding the complexity of the problems on trees with maximum degree at most $5$. On the positive side, we give a linear time algorithm for $1$-caterpillars with unbounded degree, even if the backbone is a cycle, which outperforms the previous best complexity on paths and widens the class of graphs. Finally, we answer an open question regarding the complexity of Colourful Components on graphs with maximum degree at most $5$. We show that the problem is NP-complete on $5$-coloured planar graphs with maximum degree $4$, and on $12$-coloured planar graphs with maximum degree $3$. Since the problem can be solved in polynomial-time on graphs with maximum degree $2$, the results are the best possible with regard to the maximum degree.