Geometric realizations of dichotomous ordinal graphs
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866909532713648128 |
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| author | Angelini, Patrizio Cornelsen, Sabine Haase, Carolina Hoffmann, Michael Katsanou, Eleni Montecchiani, Fabrizio Steiner, Raphael Symvonis, Antonios |
| author_facet | Angelini, Patrizio Cornelsen, Sabine Haase, Carolina Hoffmann, Michael Katsanou, Eleni Montecchiani, Fabrizio Steiner, Raphael Symvonis, Antonios |
| contents | A dichotomous ordinal graph consists of an undirected graph with a partition of the edges into short and long edges. A geometric realization of a dichotomous ordinal graph $G$ in a metric space $X$ is a drawing of $G$ in $X$ in which every long edge is strictly longer than every short edge. We call a graph $G$ pandichotomous in $X$ if $G$ admits a geometric realization in $X$ for every partition of its edge set into short and long edges. We exhibit a very close relationship between the degeneracy of a graph $G$ and its pandichotomic Euclidean or spherical dimension, that is, the smallest dimension $k$ such that $G$ is pandichotomous in $\mathbb{R}^k$ or the sphere $\mathbb{S}^k$, respectively. First, every $d$-degenerate graph is pandichotomous in $\mathbb{R}^{d}$ and $\mathbb{S}^{d-1}$ and these bounds are tight for the sphere and for $\mathbb{R}^2$ and almost tight for $\mathbb{R}^d$, for $d\ge 3$. Second, every $n$-vertex graph that is pandichotomous in $\mathbb{R}^k$ has at most $μkn$ edges, for some absolute constant $μ<7.23$. This shows that the pandichotomic Euclidean dimension of any graph is linearly tied to its degeneracy and in the special cases $k\in \{1,2\}$ resolves open problems posed by Alam, Kobourov, Pupyrev, and Toeniskoetter. Further, we characterize which complete bipartite graphs are pandichotomous in $\mathbb{R}^2$: These are exactly the $K_{m,n}$ with $m\le 3$ or $m=4$ and $n\le 6$. For general bipartite graphs, we can guarantee realizations in $\mathbb{R}^2$ if the short or the long subgraph is constrained: namely if the short subgraph is outerplanar or a subgraph of a rectangular grid, or if the long subgraph forms a caterpillar. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_07361 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric realizations of dichotomous ordinal graphs Angelini, Patrizio Cornelsen, Sabine Haase, Carolina Hoffmann, Michael Katsanou, Eleni Montecchiani, Fabrizio Steiner, Raphael Symvonis, Antonios Computational Geometry Discrete Mathematics Data Structures and Algorithms Combinatorics 05C10, 05C62, 68R10, 52C99 F.2.2; I.3.5 A dichotomous ordinal graph consists of an undirected graph with a partition of the edges into short and long edges. A geometric realization of a dichotomous ordinal graph $G$ in a metric space $X$ is a drawing of $G$ in $X$ in which every long edge is strictly longer than every short edge. We call a graph $G$ pandichotomous in $X$ if $G$ admits a geometric realization in $X$ for every partition of its edge set into short and long edges. We exhibit a very close relationship between the degeneracy of a graph $G$ and its pandichotomic Euclidean or spherical dimension, that is, the smallest dimension $k$ such that $G$ is pandichotomous in $\mathbb{R}^k$ or the sphere $\mathbb{S}^k$, respectively. First, every $d$-degenerate graph is pandichotomous in $\mathbb{R}^{d}$ and $\mathbb{S}^{d-1}$ and these bounds are tight for the sphere and for $\mathbb{R}^2$ and almost tight for $\mathbb{R}^d$, for $d\ge 3$. Second, every $n$-vertex graph that is pandichotomous in $\mathbb{R}^k$ has at most $μkn$ edges, for some absolute constant $μ<7.23$. This shows that the pandichotomic Euclidean dimension of any graph is linearly tied to its degeneracy and in the special cases $k\in \{1,2\}$ resolves open problems posed by Alam, Kobourov, Pupyrev, and Toeniskoetter. Further, we characterize which complete bipartite graphs are pandichotomous in $\mathbb{R}^2$: These are exactly the $K_{m,n}$ with $m\le 3$ or $m=4$ and $n\le 6$. For general bipartite graphs, we can guarantee realizations in $\mathbb{R}^2$ if the short or the long subgraph is constrained: namely if the short subgraph is outerplanar or a subgraph of a rectangular grid, or if the long subgraph forms a caterpillar. |
| title | Geometric realizations of dichotomous ordinal graphs |
| topic | Computational Geometry Discrete Mathematics Data Structures and Algorithms Combinatorics 05C10, 05C62, 68R10, 52C99 F.2.2; I.3.5 |
| url | https://arxiv.org/abs/2503.07361 |