On algorithmic applications of sim-width and mim-width of $(H_1, H_2)$-free graphs
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2022
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866908329805086720 |
|---|---|
| author | Munaro, Andrea Yang, Shizhou |
| author_facet | Munaro, Andrea Yang, Shizhou |
| contents | Mim-width and sim-width are among the most powerful graph width parameters, with sim-width more powerful than mim-width, which is in turn more powerful than clique-width. While several $\mathsf{NP}$-hard graph problems become tractable for graph classes whose mim-width is bounded and quickly computable, no algorithmic applications of boundedness of sim-width are known. In [Kang et al., A width parameter useful for chordal and co-comparability graphs, Theoretical Computer Science, 704:1-17, 2017], it is asked whether \textsc{Independent Set} and \textsc{$3$-Colouring} are $\mathsf{NP}$-complete on graphs of sim-width at most $1$. We observe that, for each $k \in \mathbb{N}$, \textsc{List $k$-Colouring} is polynomial-time solvable for graph classes whose sim-width is bounded and quickly computable. Moreover, we show that if the same holds for \textsc{Independent Set}, then \textsc{Independent $\mathcal{H}$-Packing} is polynomial-time solvable for graph classes whose sim-width is bounded and quickly computable. This problem is a common generalisation of \textsc{Independent Set}, \textsc{Induced Matching}, \textsc{Dissociation Set} and \textsc{$k$-Separator}. We also make progress toward classifying the mim-width of $(H_1,H_2)$-free graphs in the case $H_1$ is complete or edgeless. Our results solve some open problems in [Brettell et al., Bounding the mim-width of hereditary graph classes, Journal of Graph Theory, 99(1):117-151, 2022]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_15160 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On algorithmic applications of sim-width and mim-width of $(H_1, H_2)$-free graphs Munaro, Andrea Yang, Shizhou Data Structures and Algorithms Discrete Mathematics Combinatorics 05C85, 68R10 Mim-width and sim-width are among the most powerful graph width parameters, with sim-width more powerful than mim-width, which is in turn more powerful than clique-width. While several $\mathsf{NP}$-hard graph problems become tractable for graph classes whose mim-width is bounded and quickly computable, no algorithmic applications of boundedness of sim-width are known. In [Kang et al., A width parameter useful for chordal and co-comparability graphs, Theoretical Computer Science, 704:1-17, 2017], it is asked whether \textsc{Independent Set} and \textsc{$3$-Colouring} are $\mathsf{NP}$-complete on graphs of sim-width at most $1$. We observe that, for each $k \in \mathbb{N}$, \textsc{List $k$-Colouring} is polynomial-time solvable for graph classes whose sim-width is bounded and quickly computable. Moreover, we show that if the same holds for \textsc{Independent Set}, then \textsc{Independent $\mathcal{H}$-Packing} is polynomial-time solvable for graph classes whose sim-width is bounded and quickly computable. This problem is a common generalisation of \textsc{Independent Set}, \textsc{Induced Matching}, \textsc{Dissociation Set} and \textsc{$k$-Separator}. We also make progress toward classifying the mim-width of $(H_1,H_2)$-free graphs in the case $H_1$ is complete or edgeless. Our results solve some open problems in [Brettell et al., Bounding the mim-width of hereditary graph classes, Journal of Graph Theory, 99(1):117-151, 2022]. |
| title | On algorithmic applications of sim-width and mim-width of $(H_1, H_2)$-free graphs |
| topic | Data Structures and Algorithms Discrete Mathematics Combinatorics 05C85, 68R10 |
| url | https://arxiv.org/abs/2205.15160 |