Degeneracy Of Graph Research Articles

Discrepancy and sparsity

We study the connections between the notions of combinatorial discrepancy and graph degeneracy. In particular, we prove that the maximum discrepancy over all subgraphs H of a graph G of the neighborhood set system of H is sandwiched between Ω(log⁡deg(G)) and O(deg(G)), where deg(G) denotes the degeneracy of G. We extend this result to inequalities relating weak coloring numbers and discrepancy of graph powers and deduce a new characterization of bounded expansion classes.Then we switch to a model theoretical point of view, introduce pointer structures, and study their relations to graph classes with bounded expansion. We deduce that a monotone class of graphs has bounded expansion if and only if all the set systems definable in this class have bounded hereditary discrepancy.Using known bounds on the VC-density of set systems definable in nowhere dense classes we also give a characterization of nowhere dense classes in terms of discrepancy.As consequences of our results, we obtain a corollary on the discrepancy of neighborhood set systems of edge colored graphs, a polynomial-time algorithm to compute ε-approximations of size O(1/ε) for set systems definable in bounded expansion classes, an application to clique coloring, and even the non-existence of a quantifier elimination scheme for nowhere dense classes.

Weak degeneracy of regular graphs

Motivated by the study of greedy algorithms for graph coloring, Bernshteyn and Lee introduced a generalization of graph degeneracy, which is called weak degeneracy. In this paper, we show the lower bound of the weak degeneracy for d-regular graphs is exactly ⌊d/2⌋+1, which is tight. This result refutes the conjecture of Bernshteyn and Lee on this lower bound.

Graph degeneracy and orthogonal vector representations

We apply a technique of Sinkovic and van der Holst for constructing orthogonal vector representations of a graph whose complement has given treewidth to graphs whose complement has given degeneracy.

Degeneracy-Based Real-Time Sub-Event Detection in Twitter Stream

In this paper, we deal with the task of sub-event detection in evolving events using posts collected from the Twitter stream. By representing a sequence of successive tweets in a short time interval as a weighted graph-of-words, we are able to identify the key moments (sub-events) that compose an event using the concept of graph degeneracy. We then select a tweet to best describe each sub-event using a simple yet effective heuristic. We evaluated our approach using humangenerated summaries containing the actual important sub-events within each event and compare it to two baseline approaches using several performance metrics such as DET curves and precision/recall performance. Extensive experiments on recent sporting event streams indicate that our approach outperforms the dominant sub-event detection methods and constructs a humanreadable event summary by aggregating the most representative tweets of each sub-event.

Method for Processing Graph Degeneracy in Dynamic Geometry Based on Domain Design

Method for Processing Graph Degeneracy in Dynamic Geometry Based on Domain Design

The core decomposition of networks: theory, algorithms and applications

The core decomposition of networks has attracted significant attention due to its numerous applications in real-life problems. Simply stated, the core decomposition of a network (graph) assigns to each graph node v, an integer number c(v) (the core number), capturing how well v is connected with respect to its neighbors. This concept is strongly related to the concept of graph degeneracy, which has a long history in graph theory. Although the core decomposition concept is extremely simple, there is an enormous interest in the topic from diverse application domains, mainly because it can be used to analyze a network in a simple and concise manner by quantifying the significance of graph nodes. Therefore, there exists a respectable number of research works that either propose efficient algorithmic techniques under different settings and graph types or apply the concept to another problem or scientific area. Based on this large interest in the topic, in this survey, we perform an in-depth discussion of core decomposition, focusing mainly on: (i) the basic theory and fundamental concepts, (ii) the algorithmic techniques proposed for computing it efficiently under different settings, and (iii) the applications that can benefit significantly from it.

Sharp lower and upper bounds for the Gaussian rank of a graph

An open problem in graphical Gaussian models is to determine the smallest number of observations needed to guarantee the existence of the maximum likelihood estimator of the covariance matrix with probability one. In this paper we formulate a closely related problem in which the existence of the maximum likelihood estimator is guaranteed for all generic observations. We call the number determined by this problem the Gaussian rank of the graph representing the model. We prove that the Gaussian rank is strictly between the subgraph connectivity number and the graph degeneracy number. These bounds are sharper than the bounds known in the literature and furthermore computable in polynomial time.

CoreCluster: A Degeneracy Based Graph Clustering Framework

Graph clustering or community detection constitutes an important task forinvestigating the internal structure of graphs, with a plethora of applications in several domains. Traditional tools for graph clustering, such asspectral methods, typically suffer from high time and space complexity. In thisarticle, we present CoreCluster, an efficient graph clusteringframework based on the concept of graph degeneracy, that can be used along withany known graph clustering algorithm. Our approach capitalizes on processing thegraph in a hierarchical manner provided by its core expansion sequence, anordered partition of the graph into different levels according to the k-coredecomposition. Such a partition provides a way to process the graph inan incremental manner that preserves its clustering structure, whilemaking the execution of the chosen clustering algorithm much faster due to thesmaller size of the graph's partitions onto which the algorithm operates.

Comparing the molecular graph degeneracy of Wiener, Harary, Balaban, Randi´c and ZEP topological indices

The aim of this paper is to show that the ZEP topological index has better discrimination power than four well known topological indices in molecular chemistry: Balaban index, Harary index, Randic index, and Wiener index.

Generalized degeneracy, dynamic monopolies and maximum degenerate subgraphs

A graph G is said to be k-degenerate if any subgraph of G contains a vertex of degree at most k. The degeneracy of graphs has many applications and was widely studied in graph theory. We first generalize k-degeneracy by introducing κ-degeneracy of graphs, where κ is any non-negative function on the vertex set of the graph. We present a polynomial time algorithm to determine whether a graph is κ-degenerate. Let τ:V(G)→Z be an assignment of thresholds to the vertices of G. A subset of vertices D is said to be a τ-dynamic monopoly of G, if the vertices of G can be partitioned into subsets D0,D1,…,Dk such that D0=D and for any i∈{0,…,k−1}, each vertex v in Di+1 has at least τ(v) neighbors in D0∪⋯∪Di. The concept of dynamic monopolies is used for the formulation and analysis of spread of influence such as disease or opinion in social networks and is the subject of active research in recent years. We obtain a relationship between degeneracy and dynamic monopoly of graphs and show that these two concepts are dual of each other. Using this relationship, we introduce and study dynt(G), which is the smallest cardinality of any τ-dynamic monopoly among all threshold assignments τ with average threshold τ̄=t. We give an explicit formula for dynt(G), and obtain some lower and upper bounds for it. We show that dynt(G) is NP-complete but for complete multipartite graphs and some other classes of graphs it can be solved by polynomial time algorithms. For the regular graphs, dynt(G) can be approximated within a ratio of nearly 2. Finally we consider the problem of determining the maximum size of κ-degenerate (or k-degenerate) induced subgraphs in any graph and obtain some upper and lower bounds for the maximum size of such subgraphs.

D-cores: measuring collaboration of directed graphs based on degeneracy

Community detection and evaluation is an important task in graph mining. In many cases, a community is defined as a subgraph characterized by dense connections or interactions between its nodes. A variety of measures are proposed to evaluate different quality aspects of such communities—in most cases ignoring the directed nature of edges. In this paper, we introduce novel metrics for evaluating the collaborative nature of directed graphs—a property not captured by the single node metrics or by other established community evaluation metrics. In order to accomplish this objective, we capitalize on the concept of graph degeneracy and define a novel D-core framework, extending the classic graph-theoretic notion of \(k\)-cores for undirected graphs to directed ones. Based on the D-core, which essentially can be seen as a measure of the robustness of a community under degeneracy, we devise a wealth of novel metrics used to evaluate graph collaboration features of directed graphs. We applied the D-core approach on large synthetic and real-world graphs such as Wikipedia, DBLP, and ArXiv and report interesting results at the graph as well at the node level.

Degrees in Oriented Hypergraphs and Ramsey p-Chromatic Number

The family $D(k,m)$ of graphs having an orientation such that for every vertex $v \in V(G)$ either (outdegree) $\deg^+(v) \le k$ or (indegree) $\deg^-(v) \le m$ have been investigated recently in several papers because of the role $D(k,m)$ plays in the efforts to estimate the maximum directed cut in digraphs and the minimum cover of digraphs by directed cuts. Results concerning the chromatic number of graphs in the family $D(k,m)$ have been obtained via the notion of $d$-degeneracy of graphs. In this paper we consider a far reaching generalization of the family $D(k,m)$, in a complementary form, into the context of $r$-uniform hypergraphs, using a generalization of Hakimi's theorem to $r$-uniform hypergraphs and by showing some tight connections with the well known Ramsey numbers for hypergraphs.

A note on the cover degeneracy of graphs

AbstractWe give a 4‐chromatic planar graph, which admits a vertex partition into three parts such that the union of every two of them induces a forest. This solves a problem posed by Böhme. Also, by constructing an infinite sequence of graphs, we show that the cover degeneracy can be arbitrarily less than the chromatic number. © 2005 Wiley Periodicals, Inc. J Graph Theory

Colorings and homomorphisms of degenerate and bounded degree graphs

We are interested here in homomorphisms of undirected graphs and relate them to graph degeneracy and bounded degree property. Our main result reads that both in the abundance of counterexamples and from the complexity point of view the Brooks's theorem and first fit algorithm are the ‘only’ easy cases.

Mayer's theory analysis of repulsive and weak, long range potentials

The irreducible integrals defined in the Mayer theory of real gases are derived for one-dimensional systems having hard rod, weak long range, Curie-Weiss and van der Waals potential functions. In two cases, independent knowledge of the exact equation of state of the gas permits the derivation of general star graph degeneracy relationships. These, in turn, are useful for the study of new potential functions; a penetrable repulsive force is shown to produce attractive force behavior in the gas state equation.