Degree-constrained 2-partitions of graphs

01/18/2018

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by   Joergen Bang-Jensen, et al.

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A (δ≥ k_1,δ≥ k_2)-partition of a graph G is a vertex-partition (V_1,V_2) of G satisfying that δ(G[V_i])≥ k_i for i=1,2. We determine, for all positive integers k_1,k_2, the complexity of deciding whether a given graph has a (δ≥ k_1,δ≥ k_2)-partition. We also address the problem of finding a function g(k_1,k_2) such that the (δ≥ k_1,δ≥ k_2)-partition problem is NP-complete for the class of graphs of minimum degree less than g(k_1,k_2) and polynomial for all graphs with minimum degree at least g(k_1,k_2). We prove that g(1,k)=k for k> 3, that g(2,2)=3 and that g(2,3), if it exists, has value 4 or 5.