Marcello's completion of graphs

TL;DR

This paper introduces the concept of Marcello's completion, a new graph optimization process that iteratively adds edges to complete a graph, and defines the Marcello number as the minimal steps needed for full completion.

Contribution

It defines the Marcello completion process and the Marcello number, providing initial results and formalizing a new graph optimization problem.

Findings

Defined the Marcello number for graphs

Established initial properties of Marcello's completion

Presented foundational results on graph completion process

Abstract

This paper initiates a study on a new optimization problem with regards to graph completion. The defined procedure is called, \emph{Marcello's completion} of a graph. For graph $G$ of order $n$ the \emph{Marcello number} is obtained by iteratively constructing graphs, $G_1,G_2,\dots,G_k$ by adding a maximal number of edges between pairs of distinct, non-adjacent vertices in accordance with the \emph{Marcello rule}. If for smallest $k$ the resultant graph $G_k \cong K_n$ then the Marcello number of a graph $G$ denoted by $\varpi(G)$ is equal to $\varpi(G) = k$. By convention $\varpi(K_n) = 0$, $n \geq 1$. Certain introductory results are presented.