6,662 research outputs found
Inverse monoids of partial graph automorphisms
A partial automorphism of a finite graph is an isomorphism between its vertex
induced subgraphs. The set of all partial automorphisms of a given finite graph
forms an inverse monoid under composition (of partial maps). We describe the
algebraic structure of such inverse monoids by the means of the standard tools
of inverse semigroup theory, namely Green's relations and some properties of
the natural partial order, and give a characterization of inverse monoids which
arise as inverse monoids of partial graph automorphisms. We extend our results
to digraphs and edge-colored digraphs as well
Model Checking Lower Bounds for Simple Graphs
A well-known result by Frick and Grohe shows that deciding FO logic on trees
involves a parameter dependence that is a tower of exponentials. Though this
lower bound is tight for Courcelle's theorem, it has been evaded by a series of
recent meta-theorems for other graph classes. Here we provide some additional
non-elementary lower bound results, which are in some senses stronger. Our goal
is to explain common traits in these recent meta-theorems and identify barriers
to further progress. More specifically, first, we show that on the class of
threshold graphs, and therefore also on any union and complement-closed class,
there is no model-checking algorithm with elementary parameter dependence even
for FO logic. Second, we show that there is no model-checking algorithm with
elementary parameter dependence for MSO logic even restricted to paths (or
equivalently to unary strings), unless E=NE. As a corollary, we resolve an open
problem on the complexity of MSO model-checking on graphs of bounded max-leaf
number. Finally, we look at MSO on the class of colored trees of depth d. We
show that, assuming the ETH, for every fixed d>=1 at least d+1 levels of
exponentiation are necessary for this problem, thus showing that the (d+1)-fold
exponential algorithm recently given by Gajarsk\`{y} and Hlin\u{e}n\`{y} is
essentially optimal
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