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

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.