1 research outputs found
Almost all friendly matrices have many obstructions
A symmetric matrix with entries taken from
gives rise to a graph partition problem, asking whether a graph can be
partitioned into vertex sets matched to the rows (and corresponding
columns) of such that, if , then any two vertices between the
corresponding vertex sets are joined by an edge, and if then any two
vertices between the corresponding vertex sets are not joined by an edge. The
entry places no restriction on the edges between the corresponding sets.
This problem generalises graph colouring and graph homomorphism problems.
A graph with no -partition but such that every proper subgraph does have
an -partition is called a minimal obstruction. Feder, Hell and Xie have
defined friendly matrices and shown that non-friendly matrices have infinitely
many minimal obstructions. They showed through examples that friendly matrices
can have finitely or infinitely many minimal obstructions and gave an example
of a friendly matrix with an NP-hard partition problem. Here we show that
almost all friendly matrices have infinitely many minimal obstructions and an
NP-hard partition problem.Comment: 11 page