Jones' conjecture for Halin graphs and a bit more

TL;DR

Jones' conjecture for planar graphs is proved for the class of based planar graphs, which includes Halin graphs, by showing that any such graph $G$ satisfies $fvs(G) \le 2 \cdot cp(G)$, equivalently enabling a forest after deleting at most $2k$ vertices when $cp(G)=k$. The proof relies on an induction on the number of vertices, coupled with reductions that preserve the cycle packing and feedback vertex set sizes and a key 'good triangle' lemma on the outer face to produce a smaller graph $G'$ with strictly smaller $cp(G)$. A stronger bound, $fvs(G) \le 2 \cdot fp(G)$, is established as well. The paper highlights limitations of the based-face structure and points to open generalizations to broader planar graphs and Hamiltonian planar graphs as future directions.

Abstract

We prove Jones' famous conjecture for Halin graphs and a somewhat more general class of graphs too. A based planar graph is a planar one that has a face adjacent to every other face. We confirm Jones' conjecture for based planar graphs. Namely, if a based planar graph does not contain $k+1$ vertex-disjoint cycles, then it suffices to delete $2k$ vertices to make it acyclic.

Theorems & Definitions (3)

• proof : Proof of Claim \ref{['degree3']}
• proof : Proof of Lemma \ref{['triangle']}
• proof : Proof of Theorem \ref{['main']}