Profile and neighbourhood complexity of graphs excluding a minor and tree-structured graphs
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866915671843012608 |
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| author | Beaudou, Laurent Bok, Jan Foucaud, Florent Quiroz, Daniel A. Raymond, Jean-Florent |
| author_facet | Beaudou, Laurent Bok, Jan Foucaud, Florent Quiroz, Daniel A. Raymond, Jean-Florent |
| contents | The \emph{$r$-neighbourhood complexity} of a graph $G$ is the function counting, for a given integer $k$, the largest possible number, over all vertex-subsets $A$ of size $k$, of subsets of $A$ realized as the intersection between the $r$-neighbourhood of some vertex and $A$. A~refinement of this notion is the \emph{$r$-profile complexity}, that counts the maximum number of distinct distance-vectors from any vertex to the vertices of $A$, ignoring distances larger than~$r$. Typically, in structured graph classes such as graphs of bounded VC-dimension or chordal graphs, these functions are bounded, leading to insights into their structural properties and efficient algorithms.
We improve existing bounds on the $r$-profile complexity (and thus on the $r$-neighbourhood complexity) for graphs in several structured graph classes. We show that the $r$-profile complexity of graphs excluding $K_h$ as a minor is in $O_h(r^{3h-3}k)$. For graphs of treewidth at most~$t$, we give a bound in $O_t(r^{t+1}k)$, which is tight up to a function of~$t$ as a factor. These bounds improve results of Joret and Rambaud and answer a question of their paper [Combinatorica, 2024]. We also apply our methods to other classes of bounded expansion such as graphs excluding a fixed complete graph as a subdivision.
For outerplanar graphs, we can improve our treewidth bound by a factor of $r$ and conjecture that a similar improvement holds for graphs with bounded simple treewidth. For graphs of treelength at most~$\ell$, we give the upper bound of $O(k(r^2(\ell+1)^k))$, which we improve to $O\left (k\cdot (r 2^k + r^2k^2) \right)$ in the case of chordal graphs and $O(k^2r)$ for interval graphs.
Our bounds also imply relations between the order, diameter and metric dimension of graphs in these classes, improving results from [Beaudou et al., SIDMA 2017]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_08895 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Profile and neighbourhood complexity of graphs excluding a minor and tree-structured graphs Beaudou, Laurent Bok, Jan Foucaud, Florent Quiroz, Daniel A. Raymond, Jean-Florent Discrete Mathematics Combinatorics The \emph{$r$-neighbourhood complexity} of a graph $G$ is the function counting, for a given integer $k$, the largest possible number, over all vertex-subsets $A$ of size $k$, of subsets of $A$ realized as the intersection between the $r$-neighbourhood of some vertex and $A$. A~refinement of this notion is the \emph{$r$-profile complexity}, that counts the maximum number of distinct distance-vectors from any vertex to the vertices of $A$, ignoring distances larger than~$r$. Typically, in structured graph classes such as graphs of bounded VC-dimension or chordal graphs, these functions are bounded, leading to insights into their structural properties and efficient algorithms. We improve existing bounds on the $r$-profile complexity (and thus on the $r$-neighbourhood complexity) for graphs in several structured graph classes. We show that the $r$-profile complexity of graphs excluding $K_h$ as a minor is in $O_h(r^{3h-3}k)$. For graphs of treewidth at most~$t$, we give a bound in $O_t(r^{t+1}k)$, which is tight up to a function of~$t$ as a factor. These bounds improve results of Joret and Rambaud and answer a question of their paper [Combinatorica, 2024]. We also apply our methods to other classes of bounded expansion such as graphs excluding a fixed complete graph as a subdivision. For outerplanar graphs, we can improve our treewidth bound by a factor of $r$ and conjecture that a similar improvement holds for graphs with bounded simple treewidth. For graphs of treelength at most~$\ell$, we give the upper bound of $O(k(r^2(\ell+1)^k))$, which we improve to $O\left (k\cdot (r 2^k + r^2k^2) \right)$ in the case of chordal graphs and $O(k^2r)$ for interval graphs. Our bounds also imply relations between the order, diameter and metric dimension of graphs in these classes, improving results from [Beaudou et al., SIDMA 2017]. |
| title | Profile and neighbourhood complexity of graphs excluding a minor and tree-structured graphs |
| topic | Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2501.08895 |