Related papers: The computational strength of matchings in countab…
A graph $H$ is single-crossing if it can be drawn in the plane with at most one crossing. For any single-crossing graph $H$, we give an $O(n^4)$ time algorithm for counting perfect matchings in graphs excluding $H$ as a minor. The runtime…
We consider the problem of graph matchability in non-identically distributed networks. In a general class of edge-independent networks, we demonstrate that graph matchability can be lost with high probability when matching the networks…
Is perfect matching in NC? That is, is there a deterministic fast parallel algorithm for it? This has been an outstanding open question in theoretical computer science for over three decades, ever since the discovery of RNC matching…
We leverage an algorithm of Deming [R.W. Deming, Independence numbers of graphs -- an extension of the Koenig-Egervary theorem, Discrete Math., 27(1979), no. 1, 23--33; MR534950] to decompose a matchable graph into subgraphs with a precise…
The correspondence between weighted undirected graphs and reversible Markov chains via vertex random walks is simple and well known. Leveraging this correspondence and ideas from the theory of dynamical systems, we study the structural…
The program Reverse Mathematics in the foundations of mathematics seeks to identify the minimal axioms required to prove theorems of ordinary mathematics. One always assumes the base theory, a logical system embodying computable…
Let $G$ be a simple graph with $2n$ vertices and a perfect matching. We denote by $f(G)$ and $F(G)$ the minimum and maximum forcing number of $G$, respectively. Hetyei obtained that the maximum number of edges of graphs $G$ with a unique…
Let $t$ be a positive real number. A graph is called $t$-tough if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components. The toughness of a graph is the largest $t$ for which the graph is $t$-tough.…
Many conjectures and open problems in graph theory can either be reduced to cubic graphs or are directly stated for cubic graphs. Furthermore, it is known that for a lot of problems, a counterexample must be a snark, i.e. a bridgeless cubic…
The definition of $1$-planar graphs naturally extends graph planarity, namely a graph is $1$-planar if it can be drawn in the plane with at most one crossing per edge. Unfortunately, while testing graph planarity is solvable in linear time,…
Whether it be in normal form games, or in fair allocations, or in voter preferences in voting systems, a certain pattern of reasoning is common. From a particular profile, an agent or a group of agents may have an incentive to shift to a…
Motivated by gene set enrichment analysis, we investigate the problem of combined hypothesis testing on a graph. We introduce a general framework to effectively use the structural information of the underlying graph when testing…
We employ the mathematical programming approach in conjunction with the graph theory to study the structure of correspondent banking networks. Optimizing the network requires decisions to be made to onboard, terminate or restrict the bank…
In this paper we study conditions which guarantee the existence of perfect matchings and perfect fractional matchings in uniform hypergraphs. We reduce this problem to an old conjecture by Erd\H{o}s on estimating the maximum number of edges…
The positive zero forcing number of a graph is a graph parameter that arises from a non-traditional type of graph colouring, and is related to a more conventional version of zero forcing. We establish a relation between the zero forcing and…
The rankability of data is a recently proposed problem that considers the ability of a dataset, represented as a graph, to produce a meaningful ranking of the items it contains. To study this concept, a number of rankability measures have…
In this paper, we present two main results. First, by only one conjecture (Conjecture 2.9) for recognizing a vertex symmetric graph, which is the hardest task for our problem, we construct an algorithm for finding an isomorphism between two…
It is proved that for $n \geq 6$, the number of perfect matchings in a simple connected cubic graph on $2n$ vertices is at most $4 f_{n-1}$, with $f_n$ being the $n$-th Fibonacci number. The unique extremal graph is characterized as well.…
Drawing on some recent results that provide the formalism necessary to definite stationarity for infinite random graphs, this paper initiates the study of statistical and learning questions pertaining to these objects. Specifically, a…
Linear stochastic transitivity is a central assumption in paired comparison models that is rarely verified in practice. Empirical violations, however, are common and can substantially affect inference and ranking. We develop a class of…