On Maximum Induced Forests of the Balanced Bipartite Graphs

TL;DR

The paper investigates the interplay between the forest number $f(G)$ and the decycling number $\nabla(G)$, noting the fundamental identity $f(G)+\nabla(G)=|V(G)|$. It resolves Wang and Wu's conjecture by analyzing balanced bipartite graphs and Cartesian products, proving exact bounds and structural constraints for induced forests, and establishing a sharp minimum-degree condition $\delta(\mathcal{B}) \ge \frac{n}{2}+1$ that yields $f(\mathcal{B})=n+1$. It also extends results to graph products, showing $\nabla(G_1 \square G_2) \ge α'(G_1) α'(G_2)$ and giving precise values for prisms $T\square K_2$ via the matching number. Together, these results advance the understanding of how cycle-removal and acyclic induced subgraphs behave under graph products, with broader implications for combinatorial optimization and network design.

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 $\nabla(T\,\square\, K_2) = α'(T)$, and for arbitrary graphs $G_1$ and $G_2$, it is proved that $\nabla(G_1 \,\square\, G_2) \geq α'(G_1) α'(G_2)$, where $α'$ is the matching number.

Theorems & Definitions (9)

• Conjecture 1.1
• Theorem 2.1
• Proposition 2.2
• Theorem 2.3
• Theorem 2.4
• Corollary 2.5
• Theorem 2.6
• Theorem 2.7
• Theorem 2.8