Colourful components in $k$-caterpillars and planar graphs
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2019
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| author | Chlebíková, Janka Dallard, Clément |
| author_facet | Chlebíková, Janka Dallard, Clément |
| contents | 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]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1902_11191 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Colourful components in $k$-caterpillars and planar graphs Chlebíková, Janka Dallard, Clément Discrete Mathematics 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]. |
| title | Colourful components in $k$-caterpillars and planar graphs |
| topic | Discrete Mathematics |
| url | https://arxiv.org/abs/1902.11191 |