16 research outputs found
Stable graphs of bounded twin-width
We prove that every class of graphs that is monadically stable
and has bounded twin-width can be transduced from some class with bounded
sparse twin-width. This generalizes analogous results for classes of bounded
linear cliquewidth and of bounded cliquewidth. It also implies that monadically
stable classes of bounded twin-widthare linearly -bounded.Comment: 44 pages, 2 figure
Sparse Graphs of Twin-width 2 Have Bounded Tree-width
Twin-width is a structural width parameter introduced by Bonnet, Kim,
Thomass\'e and Watrigant [FOCS 2020]. Very briefly, its essence is a gradual
reduction (a contraction sequence) of the given graph down to a single vertex
while maintaining limited difference of neighbourhoods of the vertices, and it
can be seen as widely generalizing several other traditional structural
parameters. Having such a sequence at hand allows to solve many otherwise hard
problems efficiently. Our paper focuses on a comparison of twin-width to the
more traditional tree-width on sparse graphs. Namely, we prove that if a graph
of twin-width at most contains no subgraph for some integer
, then the tree-width of is bounded by a polynomial function of . As
a consequence, for any sparse graph class we obtain a polynomial
time algorithm which for any input graph either outputs a
contraction sequence of width at most (where depends only on
), or correctly outputs that has twin-width more than . On
the other hand, we present an easy example of a graph class of twin-width
with unbounded tree-width, showing that our result cannot be extended to higher
values of twin-width
Successor-Invariant First-Order Logic on Classes of Bounded Degree
We study the expressive power of successor-invariant first-order logic, which
is an extension of first-order logic where the usage of an additional successor
relation on the structure is allowed, as long as the validity of formulas is
independent on the choice of a particular successor. We show that when the
degree is bounded, successor-invariant first-order logic is no more expressive
than first-order logic
Kernelization using structural parameters on sparse graph classes
We prove that graph problems with finite integer index have linear kernels on graphs of bounded expansion when parameterized by the size of a modulator to constant-treedepth graphs. For nowhere dense graph classes, our result yields almost-linear kernels. We also argue that such a linear kernelization result with a weaker parameter would fail to include some of the problems covered by our framework. We only require the problems to have FII on graphs of constant treedepth. This allows to prove linear kernels also for problems such as Longest-Path/Cycle, Exact- s, t -Path, Treewidth, and Pathwidth, which do not have FII on general graphs
Faster Existential FO Model Checking on Posets
We prove that the model checking problem for the existential fragment offirst-order (FO) logic on partially ordered sets is fixed-parameter tractable(FPT) with respect to the formula and the width of a poset (the maximum size ofan antichain). While there is a long line of research into FO model checking ongraphs, the study of this problem on posets has been initiated just recently byBova, Ganian and Szeider (CSL-LICS 2014), who proved that the existentialfragment of FO has an FPT algorithm for a poset of fixed width. We improve upontheir result in two ways: (1) the runtime of our algorithm isO(f(|{\phi}|,w).n^2) on n-element posets of width w, compared to O(g(|{\phi}|).n^{h(w)}) of Bova et al., and (2) our proofs are simpler and easier to follow.We complement this result by showing that, under a certaincomplexity-theoretical assumption, the existential FO model checking problemdoes not have a polynomial kernel.Comment: Paper as accepted to the LMCS journal. An extended abstract of an earlier version of this paper has appeared at ISAAC'14. Main changes to the previous version are improvements in the Multicoloured Clique part (Section 4
Faster Existential FO Model Checking on Posets
We prove that the model checking problem for the existential fragment of
first-order (FO) logic on partially ordered sets is fixed-parameter tractable
(FPT) with respect to the formula and the width of a poset (the maximum size of
an antichain). While there is a long line of research into FO model checking on
graphs, the study of this problem on posets has been initiated just recently by
Bova, Ganian and Szeider (CSL-LICS 2014), who proved that the existential
fragment of FO has an FPT algorithm for a poset of fixed width. We improve upon
their result in two ways: (1) the runtime of our algorithm is
O(f(|{\phi}|,w).n^2) on n-element posets of width w, compared to O(g(|{\phi}|).
n^{h(w)}) of Bova et al., and (2) our proofs are simpler and easier to follow.
We complement this result by showing that, under a certain
complexity-theoretical assumption, the existential FO model checking problem
does not have a polynomial kernel