In many network applications, dense subgraphs have proven to be extremely useful. One particular type of dense subgraph known as the k-core has received a great deal of attention. k-cores have been used in a number of important applications, including identifying important nodes, speeding up community detection, network visualization, and others. However, little work has investigated the 'skeletal' structure of the k-core, and the effect of such structures on the properties of the overall k-core and network itself. In this paper, we propose the Skeletal Core Subgraph, which describes the backbone of the k-core structure of a graph. We show how to categorize graphs based on their skeletal cores, and demonstrate how to efficiently decompose a given graph into its Skeletal Core Subgraph. We show both theoretically and experimentally the relationship between the Skeletal Core Subgraph and properties of the graph, including its core resilience.