Forbidden induced subgraphs in iterative higher order line graphs

TL;DR

This paper characterizes second order line graphs and higher order line graphs through forbidden induced subgraphs, extending Beineke's classical characterization of line graphs, and analyzes specific cases for graphs with maximum degree 3 and 4.

Contribution

It provides a new characterization of second order line graphs using forbidden induced subgraphs and offers a sufficient list for higher order line graphs, expanding classical graph theory results.

Findings

Characterization of second order line graphs via forbidden induced subgraphs.

Sufficient list of forbidden subgraphs for higher order line graphs.

Complete characterization of order line graphs for graphs with maximum degree 3 and 4.

Abstract

Let $G$ be a simple finite connected graph. The line graph $L(G)$ of graph $G$ is the graph whose vertices are the edges of $G$, where $ef \in E(L(G))$ when $e \cap f \neq \emptyset$. Iteratively, the higher order line graphs are defined inductively as $L^1(G) = L(G)$ and $L^n(G) = L(L^{n-1}(G))$ for $n \geq 2$. In [Derived graphs and digraphs, Beitrage zur Graphentheorie (Teubner, Leipzig 1968), 17--33 (1968)], Beineke characterize line graphs in terms of nine forbidden subgraphs. Inspired by this result, in this paper, we characterize second order line graphs in terms of pure forbidden induced subgraphs. We also give a sufficient list of forbidden subgraphs for a graph $G$ such that $G$ is a higher order line graph. We characterize all order line graphs of graph $G$ with $\Delta(G) = 3$ and $4$.