On Maximum Induced Forests of the Balanced Bipartite Graphs

TL;DR

This paper investigates the decycling and forest numbers of Cartesian products of graphs, proving bounds and characterizations, especially for bipartite and tree graphs, and resolving a conjecture related to maximum induced forests.

Contribution

It establishes new bounds for the decycling number of Cartesian products of graphs, characterizes equality cases, and confirms a conjecture about the forest number in bipartite graphs.

Findings

Proved that the decycling number of the Cartesian product of trees is minimized by certain structures.

Characterized cases where the bounds on the decycling number are tight.

Established lower bounds for the decycling number of Cartesian products involving arbitrary graphs.

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 $\nabla(T \,\square\, T') \ge\min\{ |V(T)|,|V(T')|\} - 1$ 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.