Lagmental Vicfred

A Tree Center Is One Vertex or One Edge by Vicfred

tags: mathematics, trees, graph distance, structural graph theory

Repeatedly deleting all leaves exposes the vertices of minimum eccentricity. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

Definitions first

A finite graph \(T=(V,E)\) is a tree when it is connected and acyclic. The unique simple path \(P_{uv}\) between two vertices makes distance and recursive decomposition especially rigid.

$$ \operatorname{ecc}(v)=\max_{u\in V}d(u,v) $$

The formulas should not be merged too early. The datum \(\mathsf D\), the conclusion \(\mathsf C\), and the bridge \(\Longrightarrow\) have three different logical jobs.

$$ Z(T)=\arg\min_v\operatorname{ecc}(v),\qquad|Z(T)|\in\{1,2\} $$

A small case in full

This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.

$$ \operatorname{rad}(T)=\left\lceil\frac{\operatorname{diam}(T)}2\right\rceil,\qquad\operatorname{diam}(T)\in\{2\operatorname{rad}(T),\,2\operatorname{rad}(T)-1\} $$

The reusable statement

The compact alignment is a local map of the argument: assumptions on the first row, consequence on the second. Any generalisation must preserve that dependency.

$$ \begin{aligned} \mathsf{D}\;&:\quad \operatorname{ecc}(v)=\max_{u\in V}d(u,v),\\[5pt] \mathsf{C}\;&:\quad Z(T)=\arg\min_v\operatorname{ecc}(v),\qquad|Z(T)|\in\{1,2\}. \end{aligned} $$

A nearby false statement

A rooted tree adds a parent relation that an unrooted tree does not possess. Statements about ancestors depend on the chosen root.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] Z(T)=\arg\min_v\operatorname{ecc}(v),\qquad|Z(T)|\in\{1,2\} \end{gathered}} $$

With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.

This article was posted on Sun 30 August 2026. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.