On well-f-coveredness of lexicographic product of graphs

TL;DR

This paper investigates the well-f-covered property of lexicographic product graphs, providing characterizations and conditions for specific cases, with examples illustrating the limitations of these conditions.

Contribution

It characterizes well-f-coveredness of lexicographic product graphs when the first component is empty and offers necessary conditions for other cases, highlighting their limitations.

Findings

Characterization when the first component is empty.

Necessary conditions for the second component being empty or nonempty.

Examples demonstrating the insufficiency of these conditions.

Abstract

A simple graph G is said to be well-f-covered, whenever any two maximal induced forest in G be of the same order. In this note, well-f-coveredness of lexicographic product of two graphs in case where the first component is empty, is characterized. In cases where the second component is empty, and the second component is nonempty, a necessary condition is given, and in each one, by an example, it is shown that the given condition is not sufficient.