Line Completion Number of Grid Graph $P_n \times P_m$

TL;DR

This paper determines the line completion number of grid graphs formed by Cartesian products of paths, expanding understanding of super line graphs and their properties in specific graph classes.

Contribution

It calculates the line completion number for grid graphs $P_n imes P_m$, providing new results for this class of graphs.

Findings

Line completion number for various grid graph cases

Characterization of super line graphs in grid structures

Extension of super line graph theory to grid graphs

Abstract

The concept of super line graph was introduced in the year 1995 by Bagga, Beineke and Varma. Given a graph with at least $r$ edges, the super line graph of index $r$, $L_r(G)$, has as its vertices the sets of $r$ edges of $G$, with two adjacent if there is an edge in one set adjacent to an edge in the other set. The line completion number $lc(G)$ of a graph $G$ is the least positive integer $r$ for which $L_r(G)$ is a complete graph. In this paper, we find the line completion number of grid graph $P_n \times P_m$ for various cases of $n$ and $m$.