Branching $k$-path vertex cover of forests
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866912804649304064 |
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| author | Makarov, Mikhail |
| author_facet | Makarov, Mikhail |
| contents | We define a set $P$ to be a branching $k$-path vertex cover of an undirected forest $F$ if all leaves and isolated vertices (vertices of degree at most $1$) of $F$ belong to $P$ and every path on $k$ vertices (of length $k-1$) contains either a branching vertex (a vertex of degree at least $3$) or a vertex belonging to $P$. We define the branching $k$-path vertex cover number of an undirected forest $F$, denoted by $ψ_b(F,k)$, to be the number of vertices in the smallest branching $k$-path vertex cover of $F$. These notions for a rooted directed forest are defined similarly, with natural adjustments. We prove the lower bound $ψ_b(F,k) \geq \frac{n+3k-1}{2k}$ for undirected forests, the lower bound $ψ_b(F,k) \geq \frac{n+k}{2k}$ for rooted directed forests, and that both of them are tight. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_02685 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Branching $k$-path vertex cover of forests Makarov, Mikhail Combinatorics Discrete Mathematics 05C70 (Primary) 05C05, 05C38 (Secondary) We define a set $P$ to be a branching $k$-path vertex cover of an undirected forest $F$ if all leaves and isolated vertices (vertices of degree at most $1$) of $F$ belong to $P$ and every path on $k$ vertices (of length $k-1$) contains either a branching vertex (a vertex of degree at least $3$) or a vertex belonging to $P$. We define the branching $k$-path vertex cover number of an undirected forest $F$, denoted by $ψ_b(F,k)$, to be the number of vertices in the smallest branching $k$-path vertex cover of $F$. These notions for a rooted directed forest are defined similarly, with natural adjustments. We prove the lower bound $ψ_b(F,k) \geq \frac{n+3k-1}{2k}$ for undirected forests, the lower bound $ψ_b(F,k) \geq \frac{n+k}{2k}$ for rooted directed forests, and that both of them are tight. |
| title | Branching $k$-path vertex cover of forests |
| topic | Combinatorics Discrete Mathematics 05C70 (Primary) 05C05, 05C38 (Secondary) |
| url | https://arxiv.org/abs/2601.02685 |