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Branching $k$-path vertex cover of forests

Mikhail Makarov

TL;DR

The paper studies branching $k$-path vertex covers on forests, defining variants for undirected forests and rooted directed forests and proving tight lower bounds on the cover number $ψ_b(F,k)$. The authors solve the directed case by induction on the number of vertices and then derive the undirected case via an orientation-based reduction, yielding two sharp bounds: $ψ_b(F,k) ≥ (n+k)/(2k)$ for rooted directed forests and $ψ_b(F,k) ≥ (n+3k-1)/(2k)$ for undirected forests. Tightness is established through explicit constructions of sequences of forests $(F_i)$ with corresponding leaf-based covers achieving equality in each bound. These results clarify how leaf constraints and branching structure interact to enforce minimum cover sizes in forest graphs, with implications for long-path coverage problems in acyclic networks.

Abstract

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.

Branching $k$-path vertex cover of forests

TL;DR

The paper studies branching -path vertex covers on forests, defining variants for undirected forests and rooted directed forests and proving tight lower bounds on the cover number . The authors solve the directed case by induction on the number of vertices and then derive the undirected case via an orientation-based reduction, yielding two sharp bounds: for rooted directed forests and for undirected forests. Tightness is established through explicit constructions of sequences of forests with corresponding leaf-based covers achieving equality in each bound. These results clarify how leaf constraints and branching structure interact to enforce minimum cover sizes in forest graphs, with implications for long-path coverage problems in acyclic networks.

Abstract

We define a set to be a branching -path vertex cover of an undirected forest if all leaves and isolated vertices (vertices of degree at most ) of belong to and every path on vertices (of length ) contains either a branching vertex (a vertex of degree at least ) or a vertex belonging to . We define the branching -path vertex cover number of an undirected forest , denoted by , to be the number of vertices in the smallest branching -path vertex cover of . These notions for a rooted directed forest are defined similarly, with natural adjustments. We prove the lower bound for undirected forests, the lower bound for rooted directed forests, and that both of them are tight.
Paper Structure (3 sections, 2 theorems)

This paper contains 3 sections, 2 theorems.

Table of Contents

  1. Introduction
  2. The directed case
  3. The undirected case

Key Result

Theorem 1

Let $F$ be a rooted directed forest on $n \geq 1$ vertices. Let $k \geq 2$ be a natural number. Then $\psi_b(F,k) \geq \frac{n+k}{2k}$. That lower bound is tight (when the expression in the lower bound is an integer).

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof