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Kinetically constrained spin systems are toy models of supercooled liquids and amorphous solids. In this Perspective, we revisit the prototypical Fredrickson-Andersen (FA) kinetically constrained model from the viewpoint of K-core combinatorial optimization. Each kinetic cluster of the FA system, containing all the mutually visitable microscopic occupation configurations, is exactly the solution space of a specific instance of the K-core attack problem. The whole set of different jammed occupation patterns of the FA system is the configuration space of an equilibrium K-core problem. Based on recent theoretical results achieved on the K-core attack and equilibrium K-core problems, we discuss the thermodynamic spin glass phase transitions and the maximum occupation density of the fully unfrozen FA kinetic cluster, and the minimum occupation density and extreme vulnerability of the partially frozen (jammed) kinetic clusters. The equivalence between K-core attack and the fully unfrozen FA kinetic cluster also implies a new way of sampling K-core attack solutions., Comment: 14 pages, manuscript under review