Comparing Width Parameters on Graph Classes
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910915725623296 |
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| author | Brettell, Nick Munaro, Andrea Paulusma, Daniël Yang, Shizhou |
| author_facet | Brettell, Nick Munaro, Andrea Paulusma, Daniël Yang, Shizhou |
| contents | We study how the relationship between non-equivalent width parameters changes once we restrict to some special graph class. As width parameters, we consider treewidth, clique-width, twin-width, mim-width, sim-width and tree-independence number, whereas as graph classes we consider $K_{t,t}$-subgraph-free graphs, line graphs and their common superclass, for $t \geq 3$, of $K_{t,t}$-free graphs.
We first provide a complete comparison when restricted to $K_{t,t}$-subgraph-free graphs, showing in particular that treewidth, clique-width, mim-width, sim-width and tree-independence number are all equivalent. This extends a result of Gurski and Wanke (2000) stating that treewidth and clique-width are equivalent for the class of $K_{t,t}$-subgraph-free graphs.
Next, we provide a complete comparison when restricted to line graphs, showing in particular that, on any class of line graphs, clique-width, mim-width, sim-width and tree-independence number are all equivalent, and bounded if and only if the class of root graphs has bounded treewidth. This extends a result of Gurski and Wanke (2007) stating that a class of graphs ${\cal G}$ has bounded treewidth if and only if the class of line graphs of graphs in ${\cal G}$ has bounded clique-width.
We then provide an almost-complete comparison for $K_{t,t}$-free graphs, leaving one missing case. Our main result is that $K_{t,t}$-free graphs of bounded mim-width have bounded tree-independence number. This result has structural and algorithmic consequences. In particular, it proves a special case of a conjecture of Dallard, Milanič and Štorgel.
Finally, we consider the question of whether boundedness of a certain width parameter is preserved under graph powers. We show that the question has a positive answer for sim-width precisely in the case of odd powers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_05817 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Comparing Width Parameters on Graph Classes Brettell, Nick Munaro, Andrea Paulusma, Daniël Yang, Shizhou Combinatorics Discrete Mathematics We study how the relationship between non-equivalent width parameters changes once we restrict to some special graph class. As width parameters, we consider treewidth, clique-width, twin-width, mim-width, sim-width and tree-independence number, whereas as graph classes we consider $K_{t,t}$-subgraph-free graphs, line graphs and their common superclass, for $t \geq 3$, of $K_{t,t}$-free graphs. We first provide a complete comparison when restricted to $K_{t,t}$-subgraph-free graphs, showing in particular that treewidth, clique-width, mim-width, sim-width and tree-independence number are all equivalent. This extends a result of Gurski and Wanke (2000) stating that treewidth and clique-width are equivalent for the class of $K_{t,t}$-subgraph-free graphs. Next, we provide a complete comparison when restricted to line graphs, showing in particular that, on any class of line graphs, clique-width, mim-width, sim-width and tree-independence number are all equivalent, and bounded if and only if the class of root graphs has bounded treewidth. This extends a result of Gurski and Wanke (2007) stating that a class of graphs ${\cal G}$ has bounded treewidth if and only if the class of line graphs of graphs in ${\cal G}$ has bounded clique-width. We then provide an almost-complete comparison for $K_{t,t}$-free graphs, leaving one missing case. Our main result is that $K_{t,t}$-free graphs of bounded mim-width have bounded tree-independence number. This result has structural and algorithmic consequences. In particular, it proves a special case of a conjecture of Dallard, Milanič and Štorgel. Finally, we consider the question of whether boundedness of a certain width parameter is preserved under graph powers. We show that the question has a positive answer for sim-width precisely in the case of odd powers. |
| title | Comparing Width Parameters on Graph Classes |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2308.05817 |