Moderately beyond clique-width: reduced component max-leaf and related parameters
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866911611435876352 |
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| author | Bonnet, Édouard Chang, Yeonsu Duron, Julien Geniet, Colin Kwon, O-joung |
| author_facet | Bonnet, Édouard Chang, Yeonsu Duron, Julien Geniet, Colin Kwon, O-joung |
| contents | Reduced parameters [BKW, JCTB '26; BKRT, SODA '22] are defined via contraction sequences. Based on this framework, we introduce the reduced component max-leaf, denoted by $\operatorname{cml}^\downarrow$, where component max-leaf is the maximum number of leaves in any spanning tree of any connected component. Reduced component max-leaf is strictly sandwiched between clique-width and reduced bandwidth, it is bounded in unit interval graphs, and unbounded in planar graphs. We design polynomial-time algorithms for problems such as \textsc{Maximum Induced $d$-Regular Subgraph} and \textsc{Induced Disjoint Paths} in graphs given with a contraction sequence witnessing low $\operatorname{cml}^\downarrow$, unifying and extending tractability results for classes of bounded clique-width and unit interval graphs.
We get the following collapses in sparse classes of bounded $\operatorname{cml}^\downarrow$: bounded maximum degree implies bounded treewidth, whereas $K_{t,t}$-subgraph-freeness implies strongly sublinear treewidth; we show the latter, more generally, for classes of bounded reduced cutwidth. We establish the former result by showing that graphs with bounded $\operatorname{cml}^\downarrow$ admit balanced separators dominated by a bounded number of vertices. We then showcase an application of the reduced parameters to establishing non-transducibility results. We prove that for most reduced parameters $p^\downarrow$ (including reduced bandwidth), the family of classes of bounded $p^\downarrow$ is closed under first-order transductions. We then answer a question of [BKW '26] by showing that the 3-dimensional grids have unbounded reduced bandwidth. As the class of planar graphs (or any class of bounded genus) has bounded reduced bandwidth [BKW '26], this reproves a recent result [GPP, LICS '25] that planar graphs do not first-order transduce the 3-dimensional grids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_19138 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Moderately beyond clique-width: reduced component max-leaf and related parameters Bonnet, Édouard Chang, Yeonsu Duron, Julien Geniet, Colin Kwon, O-joung Data Structures and Algorithms Discrete Mathematics Combinatorics Reduced parameters [BKW, JCTB '26; BKRT, SODA '22] are defined via contraction sequences. Based on this framework, we introduce the reduced component max-leaf, denoted by $\operatorname{cml}^\downarrow$, where component max-leaf is the maximum number of leaves in any spanning tree of any connected component. Reduced component max-leaf is strictly sandwiched between clique-width and reduced bandwidth, it is bounded in unit interval graphs, and unbounded in planar graphs. We design polynomial-time algorithms for problems such as \textsc{Maximum Induced $d$-Regular Subgraph} and \textsc{Induced Disjoint Paths} in graphs given with a contraction sequence witnessing low $\operatorname{cml}^\downarrow$, unifying and extending tractability results for classes of bounded clique-width and unit interval graphs. We get the following collapses in sparse classes of bounded $\operatorname{cml}^\downarrow$: bounded maximum degree implies bounded treewidth, whereas $K_{t,t}$-subgraph-freeness implies strongly sublinear treewidth; we show the latter, more generally, for classes of bounded reduced cutwidth. We establish the former result by showing that graphs with bounded $\operatorname{cml}^\downarrow$ admit balanced separators dominated by a bounded number of vertices. We then showcase an application of the reduced parameters to establishing non-transducibility results. We prove that for most reduced parameters $p^\downarrow$ (including reduced bandwidth), the family of classes of bounded $p^\downarrow$ is closed under first-order transductions. We then answer a question of [BKW '26] by showing that the 3-dimensional grids have unbounded reduced bandwidth. As the class of planar graphs (or any class of bounded genus) has bounded reduced bandwidth [BKW '26], this reproves a recent result [GPP, LICS '25] that planar graphs do not first-order transduce the 3-dimensional grids. |
| title | Moderately beyond clique-width: reduced component max-leaf and related parameters |
| topic | Data Structures and Algorithms Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2604.19138 |