Degree-Constrained 2-Partitions Of Graphs

A (delta >= k(1), delta >= k(2))-partition of a graph G is a vertex-partition (V-1, V-2) of G into two nonempty sets satisfying that delta(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 (delta >= k(1), delta >= k(2))-partition.We also address the problem of finding a function g(k(1), k(2)) such that the (delta >= k(1), delta >= k(2))-partition problem is NP-complete for the class of graphs of minimum degree less than g(k(1), k(2)) and polynomial time solvable for all graphs with minimum degree at least g(k(1), k(2)). We prove that g(1, k) exists and has value k for all k >= 3, that g(2, 2) also exists and has value 3 and that g(2, 3), if it exists, has value 4 or 5. (C) 2019 Elsevier B.V. All rights reserved.