Colourful components in $k$-caterpillars and planar graphs

TL;DR

The paper studies Colourful Components and Colourful Partition on vertex-coloured graphs, with a focus on trees via $k$-caterpillars and on planar graphs. It proves NP-completeness for several $k$-caterpillar configurations with bounded maximum degree and provides linear-time algorithms for $1$-caterpillars, using strategies based on pseudocaterpillars, necklace graphs, and circular-arc graph clique covers. It derives two complexity dichotomies on trees—one by maximum degree and one by the smallest $k$ for which the input is a $k$-caterpillar—and extends hardness to certain planar, coloured graphs, answering open questions of Bulteau et al. The results consolidate the tractability frontier for partitioning into colourful components in structured graphs.

Abstract

A connected component of a vertex-coloured graph is said to be colourful if all its vertices have different colours. By extension, a graph is colourful if all its connected components are colourful. Given a vertex-coloured graph $G$ and an integer $p$, the Colourful Components problem asks whether there exist at most $p$ edges whose removal makes $G$ colourful and the Colourful Partition problem asks whether there exists a partition of $G$ into at most $p$ colourful components. In order to refine our understanding of the complexity of the problems on trees, we study both problems on $k$-caterpillars, which are trees with a central path $P$ such that every vertex not in $P$ is within distance $k$ from a vertex in $P$. We prove that Colourful Components and Colourful Partition are NP-complete on $4$-caterpillars with maximum degree $3$, $3$-caterpillars with maximum degree $4$ and $2$-caterpillars with maximum degree $5$. On the other hand, we show that the problems are linear-time solvable on $1$-caterpillars. Hence, our results imply two complexity dichotomies on trees: Colourful Components and Colourful Partition are linear-time solvable on trees with maximum degree $d$ if $d \leq 2$ (that is, on paths), and NP-complete otherwise; Colourful Components and Colourful Partition are linear-time solvable on $k$-caterpillars if $k \leq 1$, and NP-complete otherwise. We leave three open cases which, if solved, would provide a complexity dichotomy for both problems on $k$-caterpillars, for every non-negative integer $k$, with respect to the maximum degree. We also show that Colourful Components is NP-complete on $5$-coloured planar graphs with maximum degree $4$ and on $12$-coloured planar graphs with maximum degree $3$. Our results answer two open questions of Bulteau et al. mentioned in [30th Annual Symposium on Combinatorial Pattern Matching, 2019].

Figures (6)

• Figure 1: A vertex-coloured graph. The dashed parts represent a partition into colourful components. Removing the three edges between the dashed parts yields a colourful graph.
• Figure 2: An example of variable and clause gadgets used in \ref{['from phi to tree']} given a $3{,}3$-SAT instance $\phi$. The variable $x_1$ appears twice as a positive literal (in $C_2$ and $C_5$) and once as a negative literal (in $C_3$) in $\phi$. The clause $C_2$ (of size $3$) in $\phi$, which can be represented with a clause gadget of type $B$ or $C$, contains $x_1$ as a positive literal (pink vertex corresponding to vertex $x_{1,2}$). The clause $C_3$ (of size $2$) in $\phi$, which must be represented with a clause gadget of type $A$, contains $x_1$ as a negative literal (light blue vertex corresponding to vertex $\bar{x}_{1,3}$). Squared vertices have a colour that appears only once in the obtained tree $T$.
• Figure 3: A $6$/̄coloured pseudocaterpillar. Dotted edges belong to all solutions to Colourful Components (up to isomorphism) and are removed from the graph in the preprocessing.
• Figure 4: On the left, a pseudocaterpillar $G$: dotted edges are removed in the preprocessing; dashed edges are obtained from \ref{['from pseudocaterpillar to circular-arc graph']}; dotted and dashed edges form an optimal solution to Colourful Components. On the right, an arc representation of the circular-arc graph constructed from the critical bad paths in $G$ (after preprocessing): dashed segments represent a minimum clique cover and correspond to the dashed edges in $G$.
• Figure 5: A schematic representation of a necklace graph with colourful beads. The backbone is represented with plain edges and the beads with dashed ellipses.

Theorems & Definitions (33)

• Corollary 1
• Corollary 2
• Theorem 1
• proof
• Definition 1: Pseudocaterpillar
• Lemma 1
• proof
• Definition 2: $C[u]$
• Remark 1
• Definition 3: Bad path