Advances on the list stubborn problem

Authors:
Simone Dantas;Luerbio Faria;Celina M. H. de Figueiredo;Sulamita Klein;Loana T. Nogueira;Fábio Protti
Affiliations:
Universidade Federal Fluminense (UFF), Brazil;Universidade do Estado do Rio de Janeiro (UERJ), Brazil;Universidade Federal do Rio de Janeiro (UFRJ), Brazil;Universidade Federal do Rio de Janeiro (UFRJ), Brazil;Universidade Federal Fluminense (UFF), Brazil;Universidade Federal Fluminense (UFF), Brazil
Venue:
CATS '10 Proceedings of the Sixteenth Symposium on Computing: the Australasian Theory - Volume 109
Year:
2010

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Abstract

The 4-partition problem is defined as partitioning the vertex set of a graph G into at most 4 parts A, B, C, D, where each part is not required to be nonempty, and is a stable set, a clique, or has no restriction; and pairs of distinct parts are completely nonadjacent, completely adjacent, or arbitrarily adjacent. The list 4-partition problem generalizes the 4-partition problem by specifying for each vertex x, a list L(x) of parts in which x is allowed to be placed. The only list 4-partition problem not classified as either polynomial time solvable or NP-complete is the list stubborn problem (up to complementarity): A and B are stable sets, D is a clique, each vertex of A is nonadjacent to each vertex of C. We polynomially reduce the general list stubborn instance to a particular instance with a structured graph and only two types of lists. Additionally, we show that this particular list 4-partition problem is polynomially equivalent to a nonlist problem, named twofold stubborn problem.