8 research outputs found
Computational Complexity of Graph Partition under Vertex-Compaction to an Irreflexive Hexagon
In this paper, we solve a long-standing graph partition problem under vertex-compaction that has been of interest since about 1999. The graph partition problem that we consider in this paper is to decide whether or not it is possible to partition the vertices of a graph into six distinct non-empty sets A, B, C, D, E, and F, such that the vertices in each set are independent, i.e., there is no edge within any set, and an edge is possible but not necessary only between the pairs of sets A and B, B and C, C and D, D and E, E and F, and F and A, and there is no edge between any other pair of sets. We study the problem as the vertex-compaction problem for an irreflexive hexagon (6-cycle). Determining the computational complexity of this problem has been a long-standing problem of interest since about 1999, especially after the results of open problems obtained by the author on a related compaction problem appeared in 1999. We show in this paper that the vertex-compaction problem for an irreflexive hexagon is NP-complete. Our proof can be extended for larger even irreflexive cycles, showing that the vertex-compaction problem for an irreflexive even k-cycle is NP-complete, for all even k geq 6
Surjective H-Colouring over reflexive digraphs
The Surjective H-Colouring problem is to test if a given graph allows a vertex-surjective homomorphism to a fixed graph H. The complexity of this problem has been well studied for undirected (partially) reflexive graphs. We introduce endo-triviality, the property of a structure that all of its endomorphisms that do not have range of size 1 are automorphisms, as a means to obtain complexity-theoretic classifications of Surjective H-Colouring in the case of reflexive digraphs. Chen (2014) proved, in the setting of constraint satisfaction problems, that Surjective H-Colouring is NP-complete if H has the property that all of its polymorphisms are essentially unary. We give the first concrete application of his result by showing that every endo-trivial reflexive digraph H has this property. We then use the concept of endo-triviality to prove, as our main result, a dichotomy for Surjective H-Colouring when H is a reflexive tournament: if H is transitive, then Surjective H-Colouring is in NL; otherwise, it is NP-complete. By combining this result with some known and new results, we obtain a complexity classification for Surjective H-Colouring when H is a partially reflexive digraph of size at most 3
The Complexity of Approximately Counting Retractions
Let be a graph that contains an induced subgraph . A retraction from
to is a homomorphism from to that is the identity function on
. Retractions are very well-studied: Given , the complexity of deciding
whether there is a retraction from an input graph to is completely
classified, in the sense that it is known for which this problem is
tractable (assuming ). Similarly, the complexity of
(exactly) counting retractions from to is classified (assuming
). However, almost nothing is known about
approximately counting retractions. Our first contribution is to give a
complete trichotomy for approximately counting retractions to graphs of girth
at least . Our second contribution is to locate the retraction counting
problem for each in the complexity landscape of related approximate
counting problems. Interestingly, our results are in contrast to the situation
in the exact counting context. We show that the problem of approximately
counting retractions is separated both from the problem of approximately
counting homomorphisms and from the problem of approximately counting list
homomorphisms --- whereas for exact counting all three of these problems are
interreducible. We also show that the number of retractions is at least as hard
to approximate as both the number of surjective homomorphisms and the number of
compactions. In contrast, exactly counting compactions is the hardest of all of
these exact counting problems
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum