315,625 research outputs found
Weighted graphs defining facets: a connection between stable set and linear ordering polytopes
A graph is alpha-critical if its stability number increases whenever an edge
is removed from its edge set. The class of alpha-critical graphs has several
nice structural properties, most of them related to their defect which is the
number of vertices minus two times the stability number. In particular, a
remarkable result of Lov\'asz (1978) is the finite basis theorem for
alpha-critical graphs of a fixed defect. The class of alpha-critical graphs is
also of interest for at least two topics of polyhedral studies. First,
Chv\'atal (1975) shows that each alpha-critical graph induces a rank inequality
which is facet-defining for its stable set polytope. Investigating a weighted
generalization, Lipt\'ak and Lov\'asz (2000, 2001) introduce critical
facet-graphs (which again produce facet-defining inequalities for their stable
set polytopes) and they establish a finite basis theorem. Second, Koppen (1995)
describes a construction that delivers from any alpha-critical graph a
facet-defining inequality for the linear ordering polytope. Doignon, Fiorini
and Joret (2006) handle the weighted case and thus define facet-defining
graphs. Here we investigate relationships between the two weighted
generalizations of alpha-critical graphs. We show that facet-defining graphs
(for the linear ordering polytope) are obtainable from 1-critical facet-graphs
(linked with stable set polytopes). We then use this connection to derive
various results on facet-defining graphs, the most prominent one being derived
from Lipt\'ak and Lov\'asz's finite basis theorem for critical facet-graphs. At
the end of the paper we offer an alternative proof of Lov\'asz's finite basis
theorem for alpha-critical graphs
Combinatorial and Algorithmic Aspects of Monadic Stability
Nowhere dense classes of graphs are classes of sparse graphs with rich structural and algorithmic properties, however, they fail to capture even simple classes of dense graphs. Monadically stable classes, originating from model theory, generalize nowhere dense classes and close them under transductions, i.e. transformations defined by colorings and simple first-order interpretations. In this work we aim to extend some combinatorial and algorithmic properties of nowhere dense classes to monadically stable classes of finite graphs. We prove the following results.
- For every monadically stable class C and fixed integer s ? 3, the Ramsey numbers R_C(s,t) are bounded from above by ?(t^{s-1-?}) for some ? > 0, improving the bound R(s,t) ? ?(t^{s-1}/(log t)^{s-1}) known for the class of all graphs and the bounds known for k-stable graphs when s ? k.
- For every monadically stable class C and every integer r, there exists ? > 0 such that every graph G ? C that contains an r-subdivision of the biclique K_{t,t} as a subgraph also contains K_{t^?,t^?} as a subgraph. This generalizes earlier results for nowhere dense graph classes.
- We obtain a stronger regularity lemma for monadically stable classes of graphs.
- Finally, we show that we can compute polynomial kernels for the independent set and dominating set problems in powers of nowhere dense classes. Formerly, only fixed-parameter tractable algorithms were known for these problems on powers of nowhere dense classes
Yard-Sale exchange on networks: Wealth sharing and wealth appropriation
Yard-Sale (YS) is a stochastic multiplicative wealth-exchange model with two
phases: a stable one where wealth is shared, and an unstable one where wealth
condenses onto one agent. YS is here studied numerically on 1d rings, 2d square
lattices, and random graphs with variable average coordination, comparing its
properties with those in mean field (MF). Equilibrium properties in the stable
phase are almost unaffected by the introduction of a network. Measurement of
decorrelation times in the stable phase allow us to determine the critical
interface with very good precision, and it turns out to be the same, for all
networks analyzed, as the one that can be analytically derived in MF. In the
unstable phase, on the other hand, dynamical as well as asymptotic properties
are strongly network-dependent. Wealth no longer condenses on a single agent,
as in MF, but onto an extensive set of agents, the properties of which depend
on the network. Connections with previous studies of coalescence of immobile
reactants are discussed, and their analytic predictions are successfully
compared with our numerical results.Comment: 10 pages, 7 figures. Submitted to JSTA
Barcode Embeddings for Metric Graphs
Stable topological invariants are a cornerstone of persistence theory and
applied topology, but their discriminative properties are often
poorly-understood. In this paper we study a rich homology-based invariant first
defined by Dey, Shi, and Wang, which we think of as embedding a metric graph in
the barcode space. We prove that this invariant is locally injective on the
space of metric graphs and globally injective on a GH-dense subset. Moreover,
we show that is globally injective on a full measure subset of metric graphs,
in the appropriate sense.Comment: The newest draft clarifies the proofs in Sections 7 and 8, and
provides improved figures therein. It also includes a results section in the
introductio
The stable set polytope of claw-free graphs with stability number at least four. I. Fuzzy antihat graphs are W-perfect
Abstract Fuzzy antihat graphs are graphs obtained as 2-clique-bond compositions of fuzzy line graphs with three different types of three-cliqued graphs. By the decomposition theorem of Chudnovsky and Seymour [2] , fuzzy antihat graphs form a large subclass of claw-free, not quasi-line graphs with stability number at least four and with no 1-joins. A graph is W -perfect if its stable set polytope is described by: nonnegativity, rank, and lifted 5-wheel inequalities. By exploiting the polyhedral properties of the 2-clique-bond composition, we prove that fuzzy antihat graphs are W -perfect and we move a crucial step towards the solution of the longstanding open question of finding an explicit linear description of the stable set polytope of claw-free graphs
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