3,792 research outputs found
Dichotomy for tree-structured trigraph list homomorphism problems
Trigraph list homomorphism problems (also known as list matrix partition
problems) have generated recent interest, partly because there are concrete
problems that are not known to be polynomial time solvable or NP-complete. Thus
while digraph list homomorphism problems enjoy dichotomy (each problem is
NP-complete or polynomial time solvable), such dichotomy is not necessarily
expected for trigraph list homomorphism problems. However, in this paper, we
identify a large class of trigraphs for which list homomorphism problems do
exhibit a dichotomy. They consist of trigraphs with a tree-like structure, and,
in particular, include all trigraphs whose underlying graphs are trees. In
fact, we show that for these tree-like trigraphs, the trigraph list
homomorphism problem is polynomially equivalent to a related digraph list
homomorphism problem. We also describe a few examples illustrating that our
conditions defining tree-like trigraphs are not unnatural, as relaxing them may
lead to harder problems
Deciding first-order properties of nowhere dense graphs
Nowhere dense graph classes, introduced by Nesetril and Ossona de Mendez,
form a large variety of classes of "sparse graphs" including the class of
planar graphs, actually all classes with excluded minors, and also bounded
degree graphs and graph classes of bounded expansion.
We show that deciding properties of graphs definable in first-order logic is
fixed-parameter tractable on nowhere dense graph classes. At least for graph
classes closed under taking subgraphs, this result is optimal: it was known
before that for all classes C of graphs closed under taking subgraphs, if
deciding first-order properties of graphs in C is fixed-parameter tractable,
then C must be nowhere dense (under a reasonable complexity theoretic
assumption).
As a by-product, we give an algorithmic construction of sparse neighbourhood
covers for nowhere dense graphs. This extends and improves previous
constructions of neighbourhood covers for graph classes with excluded minors.
At the same time, our construction is considerably simpler than those. Our
proofs are based on a new game-theoretic characterisation of nowhere dense
graphs that allows for a recursive version of locality-based algorithms on
these classes. On the logical side, we prove a "rank-preserving" version of
Gaifman's locality theorem.Comment: 30 page
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