Very well-covered graphs by Betti splittings

A very well-covered graph is an unmixed graph without isolated vertices such that the height of its edge ideal is half of the number of vertices. We study these graphs by means of Betti splittings and mapping cone constructions. We show that the cover ideals of Cohen-Macaulay very well-covered graphs are splittable. As a consequence, we compute explicitly the minimal graded free resolution of the cover ideals of such a class of graphs and prove that these graphs have homological linear quotients. Finally, we conjecture the same is true for each power of the cover ideal of a Cohen-Macaulay very well-covered graph, and settle it in the bipartite case.