  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.
