Related papers: Nested hierarchies in planar graphs
The power graph $\mathcal{P}(G)$ is a graph with group elements as vertex set and two elements are adjacent if one is a power of the other. The order supergraph $\mathcal{S}(G)$ of the power graph $\mathcal{P}(G)$ is a graph with vertex set…
We provide a novel family of generative block-models for random graphs that naturally incorporates degree distributions: the block-constrained configuration model. Block-constrained configuration models build on the generalised…
Quasi-cliques are dense incomplete subgraphs of a graph that generalize the notion of cliques. Enumerating quasi-cliques from a graph is a robust way to detect densely connected structures with applications to bio-informatics and social…
The presence of hierarchy in many real-world networks is not yet fully explained. Complex interaction networks are often coarse-grain models of vast modular networks, where tightly connected subgraphs are agglomerated into nodes for…
We prove that the cycle space of every planar finitely separable 3-connected graph $G$ is generated by some ${\rm Aut}(G)$-invariant nested set of cycles. We also discuss the situation in the case of smaller connectivity.
A nonplanar graph G is called almost-planar if for every edge e of G, at least one of G\e and G/e is planar. In 1990, Gubser characterized 3-connected almost-planar graphs in his dissertation. However, his proof is so long that only a small…
We say that a hereditary graph class $\mathcal{G}$ is \emph{clique-sparse} if there is a constant $k=k(\mathcal{G})$ such that for every graph $G\in\mathcal{G}$, every vertex of $G$ belongs to at most $k$ maximal cliques, and any maximal…
We give the analysis of the computational complexity of coalition structure generation over graphs. Given an undirected graph G = (N,E) and a valuation function v : P(N) \to R over the subsets of nodes, the problem is to find a partition of…
Hypergraphs, capable of representing high-order interactions via hyperedges, have become a powerful tool for modeling real-world biological and social systems. Inherent relationships within these real-world systems, such as the encoding…
A hierarchical structure describing the inter-relationships of species has long been a fundamental concept in systematic biology, from Linnean classification through to the more recent quest for a 'Tree of Life.' In this paper we use an…
Hypergraphs, describing networks where interactions take place among any number of units, are a natural tool to model many real-world social and biological systems. In this work we propose a principled framework to model the organization of…
In real world domains, most graphs naturally exhibit a hierarchical structure. However, data-driven graph generation is yet to effectively capture such structures. To address this, we propose a novel approach that recursively generates…
Contagion processes have been proven to fundamentally depend on the structural properties of the interaction networks conveying them. Many real networked systems are characterized by clustered substructures representing either collections…
It is known that many networks modeling real-life complex systems are small-word (large local clustering and small diameter) and scale-free (power law of the degree distribution), and very often they are also hierarchical. Although most of…
The interplay between groups and graphs have been the most famous and productive area of algebraic graph theory. In this paper, we introduce and study the graphs whose vertex set is group G such that two distinct vertices a and b having…
The generating graph $\Gamma(G)$ of a finite group $G$ has vertex set the non-identity elements of $G$, with two elements connected exactly when they generate $G$. A coclique in a graph is an empty induced subgraph, so a coclique in…
Given a graph $G$, a configuration space of $G$ can be thought of as the set of all possible configurations of "robots" which can move throughout $G$, subject to some constraints. We introduce a type of configuration space which we call…
Product structure theorems are a collection of recent results that have been used to resolve a number of longstanding open problems on planar graphs and related graph classes. One particularly useful version states that every planar graph…
This paper presents a multiscale graph construction method using both graph and signal features. Multiscale graph is a hierarchical representation of the graph, where a node at each level indicates a cluster in a finer resolution. To obtain…
Numerical simulations and observations show that galaxies are not uniformly distributed. In cosmology, the largest known structures in the universe are galaxy filaments formed from the hierarchical clustering of galaxies due to…