$k$-edge geodetic graphs

TL;DR

This paper explores the properties of $k$-edge geodetic graphs, analyzing their characteristics in various graph classes and establishing bounds for path coverings, contributing to the understanding of geodesic structures in graphs.

Contribution

It introduces the concept of $k$-edge geodetic graphs and investigates their properties in complete bipartite, tree, and product graphs, providing bounds for path coverings.

Findings

Analyzed $k$-edge geodeticity of $K_{m,n}$

Studied $k$-edge geodeticity of trees and product graphs

Provided bounds for path coverings in various graphs

Abstract

A graph $G$ is $k$-edge geodetic graph if every edge of $G$ lies in at least one geodesic of length $k$. We studied some basic properties of $k$-edge geodetic graphs. We investigated the $k$ edge-geodeticity of complete bipartite graph $K_{m,n}$ and provide the minimum number of largest fixed order path that can cover $K_{m,n}$. We also studied the $k$-edge geodeticity of tree and the product graphs like Cartesian product, Strong product, Corona product, and provide the bounds for the minimum number of the largest fixed order path that can cover the graph.