2-core of a graph ($C_2(G)$)

The 2-core is the unique largest induced subgraph with minimum degree at least two. It is obtained by repeatedly deleting vertices of degree zero or one. Every subgraph with minimum degree two survives each deletion, which proves maximality and order independence. In a connected graph with a nonempty 2-core, the deleted vertices form rooted trees attached individually to the core vertices.