Marcello's completion conjecture for connected graphs with at most one pendant vertex

Let $G$ be a non-complete, connected graph of order $n$ with at most one pendant vertex. For each vertex $v_i$ whose degree satisfies $0<\operatorname{deg}_G(v_i)<n-1$, let $\operatorname{deg}_G(v_i)$ denote its degree, let $E(G)$ be the edge set of $G$, and let $\varpi(G)$ be the number of global iterations in Marcello's completion. Then the following assertions are made:

Marcello's completion conjecture.

(a) If

$$\sum_{0<\operatorname{deg}_G(v_i)<n-1}\operatorname{deg}_G(v_i)+|E(G)|>\frac{n(n-1)}{2},$$

then $\varpi(G)=1$.

(b) If

$$\sum_{0<\operatorname{deg}_G(v_i)<n-1}\operatorname{deg}_G(v_i)+|E(G)|=\frac{n(n-1)}{2},$$

then $\varpi(G)=1$ or $2$.

The paper presents these assertions in the context of determining Marcello numbers and understanding how many Marcello edges can be found during a global iteration. The conclusion describes the general problem of guaranteeing the maximum number of Marcello edges as open, and no resolution of these assertions is supplied.