Related papers: Graph limits and hereditary properties
Given $k$ graphs $G_{1}, \ldots, G_{k}$, their intersection is the graph $(\cap_{i\in [k]}V(G_{i}), \cap_{i\in [k]}E(G_{i}))$. Given $k$ graph classes $\mathcal{G}_{1}, \ldots , \mathcal{G}_{k}$, we call the class $\{G: \forall i \in[k],…
Large graphs are sometimes studied through their degree sequences (power law or regular graphs). We study graphs that are uniformly chosen with a given degree sequence. Under mild conditions, it is shown that sequences of such graphs have…
In this paper, we study different forbidden subgraph characterizations of the prime-order element graph $\Gamma(G)$ defined on a finite group $G$. Its set of vertices is the group $G$ and two vertices $x,y \in G$ are adjacent if the order…
This note answers extremal questions like: what is the maximum number of edges in a graph of order n, which belongs to some hereditary property. The same question is answered also for the spectral radius and other similar parameters.
In this work, we introduce the \emph{interval permutation segment (IP-SEG)} model that naturally generalizes the geometric intersection models of interval and permutation graphs. We study properties of two graph classes that arise from the…
We give an explicit construction of the weak local limit of a class of preferential attachment graphs. This limit contains all local information and allows several computations that are otherwise hard, for example, joint degree…
We unify several seemingly different graph and digraph classes under one umbrella. These classes are all broadly speaking different generalizations of interval graphs, and include, in addition to interval graphs, also adjusted interval…
This work derives an upper bound on the maximum cardinality of a family of graphs on a fixed number of vertices, in which the intersection of every two graphs in that family contains a subgraph that is isomorphic to a specified graph H.…
In this paper we consider the concept of preintersection numbers of a graph. These numbers are determined by the spectrum of the adjacency matrix of the graph, and generalize the intersection numbers of a distance-regular graph. By using…
Random graphs are more and more used for modeling real world networks such as evolutionary networks of proteins. For this purpose we look at two different models and analyze how properties like connectedness and degree distributions are…
We explore what could make recognition of particular intersection-defined classes hard. We focus mainly on unit grid intersection graphs (UGIGs), i.e., intersection graphs of unit-length axis-aligned segments and grid intersection graphs…
Grid intersection graphs are the intersection graphs of vertical and horizontal segments in the plane. When the bottom and respectively left endpoints of the vertical and horizontals segments belong to a line with negative slope, the graph…
We raise some questions about graph polynomials, highlighting concepts and phenomena that may merit consideration in the development of a general theory. Our questions are mainly of three types: When do graph polynomials have reduction…
Several graph properties are characterized as the class of graphs that admit an orientation avoiding finitely many oriented structures. For instance, if $F_k$ is the set of homomorphic images of the directed path on $k+1$ vertices, then a…
We prove that the degeneracy of graphs in a hereditary class defined by a finite set S of forbidden induced subgraphs is bounded if and only if S includes a complete graph, a complete bipartite graph and a forest.
Clique-width is a well-studied graph parameter owing to its use in understanding algorithmic tractability: if the clique-width of a graph class ${\cal G}$ is bounded by a constant, a wide range of problems that are NP-complete in general…
We consider the flip-width of geometric graphs, a notion of graph width recently introduced by Toru\'nczyk. We prove that many different types of geometric graphs have unbounded flip-width. These include interval graphs, permutation graphs,…
We introduce a variation of interval graphs, called veto interval (VI) graphs. A VI graph is represented by a set of closed intervals, each containing a point called a veto mark. The edge $ab$ is in the graph if the intervals corresponding…
We prove that for any weakly convergent sequence of finite graphs with bounded vertex degrees, there exists a topological limit graphing.
A metric probability space $M$ admits thresholds if the random geometric graph on $M$ has a threshold for every monotone graph property. We connect the existence of thresholds to the uniform expansion of $M$ and prove that all standard…