158 research outputs found
Well-Quasi-Order for Permutation Graphs Omitting a Path and a Clique
We consider well-quasi-order for classes of permutation graphs which omit both a path and a clique. Our principle result is that the class of permutation graphs omitting and a clique of any size is well-quasi-ordered. This is proved by giving a structural decomposition of the corresponding permutations. We also exhibit three infinite antichains to show that the classes of permutation graphs omitting , , and are not well-quasi-ordered
Ramsey expansions of metrically homogeneous graphs
We discuss the Ramsey property, the existence of a stationary independence
relation and the coherent extension property for partial isometries (coherent
EPPA) for all classes of metrically homogeneous graphs from Cherlin's
catalogue, which is conjectured to include all such structures. We show that,
with the exception of tree-like graphs, all metric spaces in the catalogue have
precompact Ramsey expansions (or lifts) with the expansion property. With two
exceptions we can also characterise the existence of a stationary independence
relation and the coherent EPPA.
Our results can be seen as a new contribution to Ne\v{s}et\v{r}il's
classification programme of Ramsey classes and as empirical evidence of the
recent convergence in techniques employed to establish the Ramsey property, the
expansion (or lift or ordering) property, EPPA and the existence of a
stationary independence relation. At the heart of our proof is a canonical way
of completing edge-labelled graphs to metric spaces in Cherlin's classes. The
existence of such a "completion algorithm" then allows us to apply several
strong results in the areas that imply EPPA and respectively the Ramsey
property.
The main results have numerous corollaries on the automorphism groups of the
Fra\"iss\'e limits of the classes, such as amenability, unique ergodicity,
existence of universal minimal flows, ample generics, small index property,
21-Bergman property and Serre's property (FA).Comment: 57 pages, 14 figures. Extends results of arXiv:1706.00295. Minor
revisio
Robustly Self-Ordered Graphs: Constructions and Applications to Property Testing
A graph is called self-ordered (a.k.a asymmetric) if the identity
permutation is its only automorphism. Equivalently, there is a unique
isomorphism from to any graph that is isomorphic to . We say that
is robustly self-ordered if the size of the symmetric difference
between and the edge-set of the graph obtained by permuting using any
permutation is proportional to the number of non-fixed-points of
. In this work, we initiate the study of the structure, construction and
utility of robustly self-ordered graphs.
We show that robustly self-ordered bounded-degree graphs exist (in
abundance), and that they can be constructed efficiently, in a strong sense.
Specifically, given the index of a vertex in such a graph, it is possible to
find all its neighbors in polynomial-time (i.e., in time that is
poly-logarithmic in the size of the graph).
We also consider graphs of unbounded degree, seeking correspondingly
unbounded robustness parameters. We again demonstrate that such graphs (of
linear degree) exist (in abundance), and that they can be constructed
efficiently, in a strong sense. This turns out to require very different tools.
Specifically, we show that the construction of such graphs reduces to the
construction of non-malleable two-source extractors (with very weak parameters
but with some additional natural features).
We demonstrate that robustly self-ordered bounded-degree graphs are useful
towards obtaining lower bounds on the query complexity of testing graph
properties both in the bounded-degree and the dense graph models. One of the
results that we obtain, via such a reduction, is a subexponential separation
between the query complexities of testing and tolerant testing of graph
properties in the bounded-degree graph model.Comment: Slightly modified and revised version of a CCC 2021 paper that also
appeared on ECCC 27: 149 (2020
Metrically homogeneous graphs of diameter 3
We classify countable metrically homogeneous graphs of diameter 3
Twin-Width and Polynomial Kernels
We study the existence of polynomial kernels for parameterized problems without a polynomial kernel on general graphs, when restricted to graphs of bounded twin-width. It was previously observed in [Bonnet et al., ICALP\u2721] that the problem k-Independent Set allows no polynomial kernel on graph of bounded twin-width by a very simple argument, which extends to several other problems such as k-Independent Dominating Set, k-Path, k-Induced Path, k-Induced Matching. In this work, we examine the k-Dominating Set and variants of k-Vertex Cover for the existence of polynomial kernels.
As a main result, we show that k-Dominating Set does not admit a polynomial kernel on graphs of twin-width at most 4 under a standard complexity-theoretic assumption. The reduction is intricate, especially due to the effort to bring the twin-width down to 4, and it can be tweaked to work for Connected k-Dominating Set and Total k-Dominating Set with a slightly worse bound on the twin-width.
On the positive side, we obtain a simple quadratic vertex kernel for Connected k-Vertex Cover and Capacitated k-Vertex Cover on graphs of bounded twin-width. These kernels rely on that graphs of bounded twin-width have Vapnik-Chervonenkis (VC) density 1, that is, for any vertex set X, the number of distinct neighborhoods in X is at most c?|X|, where c is a constant depending only on the twin-width. Interestingly the kernel applies to any graph class of VC density 1, and does not require a witness sequence. We also present a more intricate O(k^{1.5}) vertex kernel for Connected k-Vertex Cover.
Finally we show that deciding if a graph has twin-width at most 1 can be done in polynomial time, and observe that most graph optimization/decision problems can be solved in polynomial time on graphs of twin-width at most 1
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