Polynomial Kernels for Spanning Tree with Diversity Requirements
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| Format: | Preprint |
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2026
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| _version_ | 1866915961046564864 |
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| author | Golovach, Petr A. Majumdar, Diptapriyo Saurabh, Saket |
| author_facet | Golovach, Petr A. Majumdar, Diptapriyo Saurabh, Saket |
| contents | Given a connected undirected graph $G$, a spanning tree is a subgraph $T$ of $G$ such that $V(T) = V(G)$ and $T$ is a tree. A collection of $\ell$ spanning trees $T_1,\ldots,T_\ell$ is pairwise $k$-diverse if for every $i \neq j$, $|E(T_i) \triangle E(T_j)| \geq k$. Given a connected undirected graph $G$ and integers $p, q, k, \ell$, Leaf & Internal-Constrained Diverse Spanning Trees asks whether there are $\ell$ distinct spanning trees $T_1,\ldots,T_{\ell}$ of $G$ that are pairwise $k$-diverse such that each tree has at least $p$ leaves and at least $q$ internal vertices. Similarly, Leaf & Non-terminal-Constrained Diverse Spanning Trees takes a connected undirected graph $G$, $V_{NT}\subseteq V(G)$, and three integers $p, k, \ell$, and asks if $G$ has $\ell$ spanning trees that are pairwise $k$-diverse, and each has at least $p$ leaves and conains the vertices of $V_{NT}$ as internal. We consider these two problems from the kernelization perspective and provide polynomial kernels for Leaf & Internal-Constrained Diverse Spanning Trees and Leaf & Non-terminal-Constrained Diverse Spanning Trees, when parameterized by $p + q + k + \ell$ and $p + |V_{\rm NT}| + k + \ell$, respectively. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_24571 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Polynomial Kernels for Spanning Tree with Diversity Requirements Golovach, Petr A. Majumdar, Diptapriyo Saurabh, Saket Data Structures and Algorithms Discrete Mathematics Given a connected undirected graph $G$, a spanning tree is a subgraph $T$ of $G$ such that $V(T) = V(G)$ and $T$ is a tree. A collection of $\ell$ spanning trees $T_1,\ldots,T_\ell$ is pairwise $k$-diverse if for every $i \neq j$, $|E(T_i) \triangle E(T_j)| \geq k$. Given a connected undirected graph $G$ and integers $p, q, k, \ell$, Leaf & Internal-Constrained Diverse Spanning Trees asks whether there are $\ell$ distinct spanning trees $T_1,\ldots,T_{\ell}$ of $G$ that are pairwise $k$-diverse such that each tree has at least $p$ leaves and at least $q$ internal vertices. Similarly, Leaf & Non-terminal-Constrained Diverse Spanning Trees takes a connected undirected graph $G$, $V_{NT}\subseteq V(G)$, and three integers $p, k, \ell$, and asks if $G$ has $\ell$ spanning trees that are pairwise $k$-diverse, and each has at least $p$ leaves and conains the vertices of $V_{NT}$ as internal. We consider these two problems from the kernelization perspective and provide polynomial kernels for Leaf & Internal-Constrained Diverse Spanning Trees and Leaf & Non-terminal-Constrained Diverse Spanning Trees, when parameterized by $p + q + k + \ell$ and $p + |V_{\rm NT}| + k + \ell$, respectively. |
| title | Polynomial Kernels for Spanning Tree with Diversity Requirements |
| topic | Data Structures and Algorithms Discrete Mathematics |
| url | https://arxiv.org/abs/2604.24571 |