On minimal free resolutions of the cover ideals of clique-whiskered graphs
Yuji Muta, Naoki Terai
TL;DR
This work delivers explicit constructions of minimal free resolutions for the cover ideals of clique-whiskered graphs, enabling direct computation of graded Betti numbers, projective dimension, and Castelnuovo-Mumford regularity. It extends these techniques to multi-clique-whiskered graphs, establishing vertex decomposability and sequential Cohen-Macaulayness, and derives regularity equal to the induced matching number. The results specialize to important graph classes, including Cohen-Macaulay chordal, clique corona, and Cohen-Macaulay Cameron-Walker graphs, with explicit resolutions and local cohomology Hilbert-series formulas. Overall, the paper advances explicit, combinatorially governed resolutions and invariants for cover and edge ideals across broad graph families, including very well-covered graphs.
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.