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Minimum k‐cores and the k‐core polytope

The minimum k‐core problem asks for the smallest induced subgraph of minimum degree k. It has been shown that this problem is NP‐hard, and thus sophisticated techniques are required to obtain good solutions and approximations. In this article, the minimum k‐core problem is modeled as a binary integer program and relaxed as a linear program. Since the relaxation may yield a non‐integral solution, a branch‐and‐cut framework is used to find an integral optimal solution. It is shown that the edge and cycle transversals of the graph give valid inequalities for the convex hull of the k‐core polytope—which can be further generalized to a family of ℓ ‐core transversals. Further, a heuristic for the transversal of the minimal ℓ ‐cores is given with its associated valid inequality. Additionally, improved valid inequalities are generated using bounds involving the girth of the graph. Multiple heuristics are explored for finding initial bounds for the branching process utilizing the degree distribution of the graph. Finally, numerical results are given comparing the branch‐and‐bound, branch‐and‐cut, and heuristic techniques.