Lower bounds on collective additive spanners
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arXiv
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2025
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| author | Corneil, Derek G. Dragan, Feodor F. Köhler, Ekkehard Xiang, Yang |
| author_facet | Corneil, Derek G. Dragan, Feodor F. Köhler, Ekkehard Xiang, Yang |
| contents | In this paper we present various lower bound results on collective tree spanners and on spanners of bounded treewidth. A graph $G$ is said to admit a system of $μ$ collective additive tree $c$-spanners if there is a system $\cal{T}$$(G)$ of at most $μ$ spanning trees of $G$ such that for any two vertices $u,v$ of $G$ a tree $T\in \cal{T}$$(G)$ exists such that the distance in $T$ between $u$ and $v$ is at most $c$ plus their distance in $G$. A graph $G$ is said to admit an additive $k$-treewidth $c$-spanner if there is a spanning subgraph $H$ of $G$ with treewidth $k$ such that for any pair of vertices $u$ and $v$ their distance in $H$ is at most $c$ plus their distance in $G$. Among other results, we show that:
$\bullet$ Any system of collective additive tree $1$ -- spanners must have $Ω(\sqrt[3]{\log n})$ spanning trees for some unit interval graphs;
$\bullet$ No system of a constant number of collective additive tree $2$-spanners can exist for strongly chordal graphs;
$\bullet$ No system of a constant number of collective additive tree $3$-spanners can exist for chordal graphs;
$\bullet$ No system of a constant number of collective additive tree $c$-spanners can exist for weakly chordal graphs as well as for outerplanar graphs for any constant $c\geq 0$;
$\bullet$ For any constants $k \ge 2$ and $c \ge 1$ there are graphs of treewidth $k$ such that no spanning subgraph of treewidth $k-1$ can be an additive $c$-spanner of such a graph.
All these lower bound results apply also to general graphs. Furthermore, they %results complement known upper bound results with tight lower bound results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_18508 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lower bounds on collective additive spanners Corneil, Derek G. Dragan, Feodor F. Köhler, Ekkehard Xiang, Yang Combinatorics Data Structures and Algorithms In this paper we present various lower bound results on collective tree spanners and on spanners of bounded treewidth. A graph $G$ is said to admit a system of $μ$ collective additive tree $c$-spanners if there is a system $\cal{T}$$(G)$ of at most $μ$ spanning trees of $G$ such that for any two vertices $u,v$ of $G$ a tree $T\in \cal{T}$$(G)$ exists such that the distance in $T$ between $u$ and $v$ is at most $c$ plus their distance in $G$. A graph $G$ is said to admit an additive $k$-treewidth $c$-spanner if there is a spanning subgraph $H$ of $G$ with treewidth $k$ such that for any pair of vertices $u$ and $v$ their distance in $H$ is at most $c$ plus their distance in $G$. Among other results, we show that: $\bullet$ Any system of collective additive tree $1$ -- spanners must have $Ω(\sqrt[3]{\log n})$ spanning trees for some unit interval graphs; $\bullet$ No system of a constant number of collective additive tree $2$-spanners can exist for strongly chordal graphs; $\bullet$ No system of a constant number of collective additive tree $3$-spanners can exist for chordal graphs; $\bullet$ No system of a constant number of collective additive tree $c$-spanners can exist for weakly chordal graphs as well as for outerplanar graphs for any constant $c\geq 0$; $\bullet$ For any constants $k \ge 2$ and $c \ge 1$ there are graphs of treewidth $k$ such that no spanning subgraph of treewidth $k-1$ can be an additive $c$-spanner of such a graph. All these lower bound results apply also to general graphs. Furthermore, they %results complement known upper bound results with tight lower bound results. |
| title | Lower bounds on collective additive spanners |
| topic | Combinatorics Data Structures and Algorithms |
| url | https://arxiv.org/abs/2504.18508 |