Constructing edge-disjoint Steiner trees in Cartesian product networks
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arXiv
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| Hauptverfasser: | , , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866911865644253184 |
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| author | Li, Rui Gutin, Gregory Zhang, He Wang, Zhao Zhang, Xiaoyan Mao, Yaping |
| author_facet | Li, Rui Gutin, Gregory Zhang, He Wang, Zhao Zhang, Xiaoyan Mao, Yaping |
| contents | Cartesian product networks are always regarded as a tool for ``combining'' two given networks with established properties to obtain a new one that inherits properties from both. For a graph $F=(V,E)$ and a set $S\subseteq V(F)$ of at least two vertices, \emph{an $S$-Steiner tree} or \emph{a Steiner tree connecting $S$} (or simply, \emph{an $S$-tree}) is a subgraph $T=(V',E')$ of $F$ that is a tree with $S\subseteq V'$. For $S\subseteq V(F)$ and $|S|\geq 2$, the {\it generalized local edge-connectivity} $λ(S)$ is the maximum number of edge-disjoint Steiner trees connecting $S$ in $F$. For an integer $k$ with $2\leq k\leq n$, the {\it generalized $k$-edge-connectivity} $λ_k(F)$ of a graph $F$ is defined as $λ_k(F)=\min\{λ(S)\,|\,S\subseteq V(F) \ and \ |S|=k\}$.In this paper, we give sharp upper and lower bounds for $λ_k(G\Box H)$, where $\Box$ is the Cartesian product operation, and $G,H$ are two graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_12933 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Constructing edge-disjoint Steiner trees in Cartesian product networks Li, Rui Gutin, Gregory Zhang, He Wang, Zhao Zhang, Xiaoyan Mao, Yaping Combinatorics Cartesian product networks are always regarded as a tool for ``combining'' two given networks with established properties to obtain a new one that inherits properties from both. For a graph $F=(V,E)$ and a set $S\subseteq V(F)$ of at least two vertices, \emph{an $S$-Steiner tree} or \emph{a Steiner tree connecting $S$} (or simply, \emph{an $S$-tree}) is a subgraph $T=(V',E')$ of $F$ that is a tree with $S\subseteq V'$. For $S\subseteq V(F)$ and $|S|\geq 2$, the {\it generalized local edge-connectivity} $λ(S)$ is the maximum number of edge-disjoint Steiner trees connecting $S$ in $F$. For an integer $k$ with $2\leq k\leq n$, the {\it generalized $k$-edge-connectivity} $λ_k(F)$ of a graph $F$ is defined as $λ_k(F)=\min\{λ(S)\,|\,S\subseteq V(F) \ and \ |S|=k\}$.In this paper, we give sharp upper and lower bounds for $λ_k(G\Box H)$, where $\Box$ is the Cartesian product operation, and $G,H$ are two graphs. |
| title | Constructing edge-disjoint Steiner trees in Cartesian product networks |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2301.12933 |