Algebraic Properties of Clique Complexes of Line Graphs

TL;DR

This paper explores the algebraic and topological properties of clique complexes derived from line graphs, establishing equivalences among various properties and providing efficient algorithms for their recognition and classification.

Contribution

It characterizes when clique complexes of line graphs are Cohen-Macaulay, sequentially Cohen-Macaulay, or Gorenstein, and develops linear time algorithms for these classifications.

Findings

Equivalence of sequential Cohen-Macaulay, shellable, and vertex decomposable for clique complexes.

Complete characterization of graphs with Cohen-Macaulay, sequentially Cohen-Macaulay, or Gorenstein clique complexes.

Linear time algorithms to recognize line graphs and determine their algebraic properties.

Abstract

Let $H$ be a simple undirected graph and $G=\mathrm{L}(H)$ be its line graph. Assume that $\Delta(G)$ denotes the clique complex of $G$. We show that $\Delta(G)$ is sequentially Cohen-Macaulay if and only if it is shellable if and only if it is vertex decomposable. Moreover if $\Delta(G)$ is pure, we prove that these conditions are also equivalent to being strongly connected. Furthermore, we state a complete characterizations of those $H$ for which $\Delta(G)$ is Cohen-Macaulay, sequentially Cohen-Macaulay or Gorenstein. We use these characterizations to present linear time algorithms which take a graph $G$, check whether $G$ is a line graph and if yes, decide if $\Delta(G)$ is Cohen-Macaulay or sequentially Cohen-Macaulay or Gorenstein.