Computational Complexity Relationship between Compaction, Vertex-Compaction, and Retraction

Abstract

In this paper, we show a very close relationship between the compaction, vertex-compaction, and retraction problems for reflexive and bipartite graphs. Similar to a long-standing open problem concerning whether the compaction and retraction problems are polynomially equivalent, the relationships that we present relate to our problems concerning whether the compaction and vertex-compaction problems are polynomially equivalent, and whether the vertex-compaction and retraction problems are polynomially equivalent. The relationships that we present also relate to the constraint satisfaction problem, providing evidence that similar to the compaction and retraction problems, it is also likely to be difficult to give a complete computational complexity classification of the vertex-compaction problem for every fixed reflexive or bipartite graph. In this paper, we however give a complete computational complexity classification of the vertex-compaction problem for all graphs, including even partially reflexive graphs, with four or fewer vertices, by giving proofs based on mostly just knowing the computational complexity classification results of the compaction problem for such graphs determined earlier by the author. Our results show that the compaction, vertex-compaction, and retraction problems are polynomially equivalent for every graph with four or fewer vertices.