On minimal free resolutions of the cover ideals of clique-whiskered graphs

TL;DR

This paper constructs explicit minimal free resolutions for cover ideals of clique-whiskered graphs, including several subclasses, and introduces multi-clique-whiskered graphs with properties like vertex decomposability and formulas for algebraic invariants.

Contribution

It provides explicit constructions of minimal free resolutions for cover ideals of clique-whiskered graphs and introduces multi-clique-whiskered graphs with new algebraic and combinatorial properties.

Findings

Explicit minimal free resolutions for cover ideals of clique-whiskered graphs.

Multi-clique-whiskered graphs are vertex decomposable and sequentially Cohen-Macaulay.

Formulas for projective dimension and Castelnuovo--Mumford regularity.

Abstract

We explicitly construct a minimal free resolution of the cover ideals of clique-whiskered graphs. In particular, Cohen--Macaulay chordal graphs, clique corona graphs, and Cohen--Macaulay Cameron--Walker graphs are examples of clique-whiskered graphs. We also introduce multi-clique-whiskered graphs as a generalization of both clique-whiskered graphs and multi-whisker graphs. We prove that multi-clique-whiskered graphs are vertex decomposable and hence sequentially Cohen--Macaulay. Moreover, we provide formulas for the projective dimension and the Castelnuovo--Mumford regularity of their edge ideals. Finally, we construct minimal free resolutions of the cover ideals of both multi-clique-whiskered graphs and very well-covered graphs.