6 research outputs found
Monotone graph limits and quasimonotone graphs
The recent theory of graph limits gives a powerful framework for
understanding the properties of suitable (convergent) sequences of
graphs in terms of a limiting object which may be represented by a symmetric
function on , i.e., a kernel or graphon. In this context it is
natural to wish to relate specific properties of the sequence to specific
properties of the kernel. Here we show that the kernel is monotone (i.e.,
increasing in both variables) if and only if the sequence satisfies a
`quasi-monotonicity' property defined by a certain functional tending to zero.
As a tool we prove an inequality relating the cut and norms of kernels of
the form with and monotone that may be of interest in its
own right; no such inequality holds for general kernels.Comment: 38 page
Counting Perfect Matchings and the Switch Chain
We examine the problem of exactly or approximately counting all perfect matchings in hereditary classes of nonbipartite graphs. In particular, we consider the switch Markov chain of Diaconis, Graham, and Holmes. We determine the largest hereditary class for which the chain is ergodic, and define a large new hereditary class of graphs for which it is rapidly mixing. We go on to show that the chain has exponential mixing time for a slightly larger class. We also examine the question of ergodicity of the switch chain in an arbitrary graph. Finally, we give exact counting algorithms for three classes
Interval graph limits
We work out the graph limit theory for dense interval graphs. The theory
developed departs from the usual description of a graph limit as a symmetric
function on the unit square, with and uniform on the interval
. Instead, we fix a and change the underlying distribution of the
coordinates and . We find choices such that our limits are continuous.
Connections to random interval graphs are given, including some examples. We
also show a continuity result for the chromatic number and clique number of
interval graphs. Some results on uniqueness of the limit description are given
for general graph limits.Comment: 28 pages, 4 figure
Critical properties of bipartite permutation graphs
The class of bipartite permutation graphs enjoys many nice and important properties. In particular, this class is critically important in the study of clique‐ and rank‐width of graphs, because it is one of the minimal hereditary classes of graphs of unbounded clique‐ and rank‐width. It also contains a number of important subclasses, which are critical with respect to other parameters, such as graph lettericity or shrub‐depth, and with respect to other notions, such as well‐quasi‐ordering or complexity of algorithmic problems. In the present paper we identify critical subclasses of bipartite permutation graphs of various types
Quasimonotone graphs
For any class C of bipartite graphs, we define quasi-C to be the class of all graphs G such that every bipartition of G belongs to C. This definition is motivated by a generalisation of the switch Markov chain on perfect matchings from bipartite graphs to nonbipartite graphs. The monotone graphs, also known as bipartite permutation graphs and proper interval bigraphs, are such a class of bipartite graphs. We investigate the structure of quasi-monotone graphs and hence construct a polynomial time recognition algorithm for graphs in this class