We show that the countably infinite union of infinite grids, H say, is minor-universal in the class of all graphs that can be drawn in the plane without vertex accumulation points, i.e. that H contains every such graph as a minor. Furthermore, we characterize the graphs that occur as minors of the infinite grid by a natural topological condition on their embeddings.

Numerous applications in scheduling, such as resource allocation or steel manufacturing, can be modeled using the NP-hard Independent Set problem (given an undirected graph and an integer k, find a set of at least k pairwise non-adjacent vertices). Here, one encounters special graph classes like 2-union graphs (edge-wise unions of two interval graphs) and strip graphs (edge-wise unions of an in...

Let $G$ be a graph with eigenvalues $lambda_1(G)geqcdotsgeqlambda_n(G)$. In this paper we find all simple graphs $G$ such that $G$ has at most twelve vertices and $G$ has exactly two non-negative eigenvalues. In other words we find all graphs $G$ on $n$ vertices such that $nleq12$ and $lambda_1(G)geq0$, $lambda_2(G)geq0$ and $lambda_3(G)0$, $lambda_2(G)>0$ and $lambda_3(G)

In a graph coloring, each color class induces a disjoint union of isolated vertices. A graph subcoloring generalizes this concept, since here each color class induces a disjoint union of complete graphs. Erd˝ os and, independently, Albertson et al., proved that every graph of maximum degree at most 3 has a 2-subcoloring. We point out that this fact is best possible with respect to degree constr...

A graph is polar if the vertex set can be partitioned into A and B in such a way that A induces a complete multipartite graph and B induces a disjoint union of cliques (i.e., the complement of a complete multipartite graph). Polar graphs naturally generalize several classes of graphs such as bipartite graphs, cobipartite graphs and split graphs. Recognizing polar graphs is an NP-complete proble...

Summary In this article the sum (or disjoint union) of graphs is formalized in Mizar system [4], [1], based on formalization [9].

This note attempts to understand graph limits as defined by Lovasz and Szegedy (2006)} in terms of harmonic analysis on semigroups. is done representing probability distributions random exchangeable graphs mixtures characters the semigroup unlabeled with node-disjoint union, thereby providing an alternative derivation de Finetti's theorem for graphs.

If X is a geodesic metric space and x1, x2, x3 ∈ X , a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X . The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X . We denote by δ(X) the sharpest hyperbolicity constant of X , i.e...