Related papers: Every Minor-Closed Property of Sparse Graphs is Te…
In graph property testing the task is to distinguish whether a graph satisfies a given property or is "far" from having that property, preferably with a sublinear query and time complexity. In this work we initiate the study of property…
It is known that families of graphs with a semialgebraic edge relation of bounded complexity satisfy much stronger regularity properties than arbitrary graphs, and that they can be decomposed into very homogeneous semialgebraic pieces up to…
A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by no more than one other edge (and any pair of crossing edges cross only once). A non-1-planar graph $G$ is minimal if the graph $G-e$ is 1-planar for every…
Random planar graphs have been the subject of much recent work. Many basic properties of the standard uniform random planar graph P_{n}, by which we mean a graph chosen uniformly at random from the set of all planar graphs with vertex set…
We study property testing in the \emph{random neighbor oracle} model for graphs, originally introduced by Czumaj and Sohler [STOC 2019]. Specifically, we initiate the study of characterizing the graph families that are $H$-\emph{testable}…
We give a distributed algorithm in the {\sf CONGEST} model for property testing of planarity with one-sided error in general (unbounded-degree) graphs. Following Censor-Hillel et al. (DISC 2016), who recently initiated the study of property…
A graph property P is strongly testable if for every fixed \epsilon>0 there is a one-sided \epsilon-tester for P whose query complexity is bounded by a function of \epsilon. In classifying the strongly testable graph properties, the first…
There has been substantial interest in estimating the value of a graph parameter, i.e., of a real-valued function defined on the set of finite graphs, by querying a randomly sampled substructure whose size is independent of the size of the…
Let $G$ be a finite graph with minimum degree $r$. Form a random subgraph $G_p$ of $G$ by taking each edge of $G$ into $G_p$ independently and with probability $p$. We prove that for any constant $\epsilon>0$, if $p=\frac{1+\epsilon}{r}$,…
Stability and dependence are model-theoretic notions that have recently proved highly effective in the study of structural and algorithmic properties of hereditary graph classes, and are considered key notions for generalizing to hereditary…
Given a `genus' function $g=g(n)$, we let $\mathcal{E}^g$ be the class of all graphs $G$ such that if $G$ has order $n$ (that is, has $n$ vertices) then it is embeddable in a surface of Euler genus at most $g(n)$. Let the random graph $R_n$…
We show that for various classes C of sparse graphs, and several measures of distance to such classes (such as edit distance and elimination distance), the problem of determining the distance of a given graph G to C is fixed-parameter…
We introduce a curvature function for planar graphs to study the connection between the curvature and the geometric and spectral properties of the graph. We show that non-positive curvature implies that the graph is infinite and locally…
We study the problem of recognizing the cluster structure of a graph in the framework of property testing in the bounded degree model. Given a parameter $\varepsilon$, a $d$-bounded degree graph is defined to be $(k, \phi)$-clusterable, if…
Let $\mathbb{G}^{D}$ be the set of graphs $G(V,\, E)$ with $\left|V\right|=n$, and the degree sequence equal to $D=(d_{1},\, d_{2},\,\dots,\, d_{n})$. In addition, for $\frac{1}{2}<a<1$, we define the set of graphs with an almost given…
We show that every locally sparse graph contains a linearly sized expanding subgraph. For constants $c_1>c_2>1$, $0<\alpha<1$, a graph $G$ on $n$ vertices is called a $(c_1,c_2,\alpha)$-graph if it has at least $c_1n$ edges, but every…
We investigate the threshold probability for connectivity of sparse graphs under weak assumptions. As a corollary this completely solve the problem for Cartesian powers of arbitrary graphs. In detail, let $G$ be a connected graph on $k$…
In this paper, we study the outerplanarity of planar graphs, i.e., the number of times that we must (in a planar embedding that we can initially freely choose) remove the outerface vertices until the graph is empty. It is well-known that…
We consider the following two algorithmic problems: given a graph $G$ and a subgraph $H\subseteq G$, decide whether $H$ is an isometric or a geodesically convex subgraph of $G$. It is relatively easy to see that the problems can be solved…
Reidl, S\'anchez Villaamil, and Stravopoulos (2019) characterized graph classes of bounded expansion as follows: A class $\mathcal{C}$ closed under subgraphs has bounded expansion if and only if there exists a function $f:\mathbb{N} \to…