On decycling and forest numbers of Cartesian products of trees

TL;DR

This paper investigates the decycling and forest numbers of Cartesian products of trees, proving key inequalities and characterizing cases, thereby resolving a conjecture and establishing new bounds for these graph parameters.

Contribution

It proves that the decycling number of Cartesian products of trees is minimized by stars, resolves a conjecture relating forest numbers, and establishes bounds involving matching numbers.

Findings

Decycling number of Cartesian product of trees is minimized by stars.

Resolved Wang and Wu's conjecture on forest numbers.

Established lower bounds for decycling numbers involving matching numbers.

Abstract

The decycling number $\nabla(G)$ of a graph $G$ is the minimum number of vertices that must be removed to eliminate all cycles in $G$. The forest number $f(G)$ is the maximum number of vertices that induce a forest in $G$. So $\nabla(G) + f(G) = |V(G)|$. For the Cartesian product $T \,\square\, T'$ of trees $T$ and $T'$ it is proved that $\nabla(S_n \,\square\, S_{n'}) \leq \nabla(T \,\square\, T')$, thus resolving the conjecture of Wang and Wu asserting that $f(T \,\square\, T') \leq f(S_n \,\square\, S_{n'})$. It is shown that $\nabla(T \,\square\, T') \ge |V(T)| - 1$ and the equality cases characterized. For prisms over trees, it is proved that $\nabla(T\,\square\, K_2) = \alpha'(T)$, and for arbitrary graphs $G_1$ and $G_2$, it is proved that $\nabla(G_1 \,\square\, G_2) \geq \alpha'(G_1) \alpha'(G_2)$, where $\alpha'$ is the matching number.