A construction of sequentially Cohen-Macaulay graphs

TL;DR

This paper introduces a new class of graphs called multiple clique cluster-whiskered graphs and proves they are vertex decomposable, leading to their independence complexes being sequentially Cohen-Macaulay, with detailed properties and algebraic invariants studied.

Contribution

The paper constructs the class of multiple clique cluster-whiskered graphs and proves their vertex decomposability, establishing their independence complexes as sequentially Cohen-Macaulay, and analyzes their combinatorial and algebraic properties.

Findings

All graphs $G^{md}$ are vertex decomposable.

The independence complex ${ m Ind} hinspace G^{md}$ is sequentially Cohen-Macaulay.

Betti numbers of the cover ideal $I_c(G^{md})$ are computed.

Abstract

For every simple graph $G$, a class of multiple clique cluster-whiskered graphs $G^{md}$ is introduced, and it is shown that all graphs $G^{md}$ are vertex decomposable, thus the independence simplicial complex ${\rm Ind}\,G^{md}$ is sequentially Cohen-Macaulay; the properties of the graphs $G^{md}$ and the clique-whiskered graph $G^\pi$ are studied, including the enumeration of facets of the complex ${\rm Ind}\, G^{\pi}$ and, the calculation of Betti numbers of the cover ideal $I_c(G^{md})$.