Find K-Core Components
The Wolfram System allows a graph to be hierarchically decomposed into its k-cores using KCoreComponents.
g = [image];cores = Table[KCoreComponents[g, k], {k, 1, 3}]Show[{Graphics[GraphicsComplex[GraphEmbedding[g], Table[{Hue[1 - k / (Length[cores] + 1), .3, .7], Thickness[0.14], CapForm["Round"], Line[List@@@EdgeList[Subgraph[g, Join @@ cores[[k]]]]]}, {k, 1, Length[cores]}]]], Annotate[g, { VertexStyle -> EdgeForm[{White, Opacity[1]}], EdgeStyle -> Directive[Thick, Opacity[1], White]}]}]