Asymptotic Modularity of Some Graph Classes

Given a graph $G$ and an integer $k$, two players take turns coloring the vertices of $G$ one by one using $k$ colors so that neighboring vertices get different colors. The first player wins iff at the end of the game all the vertices of $G$ are colored. The game chromatic number $\chi_g(G)$ is the minimum $k$ for which the first player has a winning strategy. The paper [T. Bohman, A. M. Frieze, and B. Sudakov, Random Structures Algorithms, 32 (2008), pp. 223--235] began the analysis of the asymptotic behavior of this parameter for a random graph $G_{n,p}$. This paper provides some further analysis for graphs with constant average degree, i.e., $np=O(1)$, and for random regular graphs. We show that with high probability (w.h.p.) $c_1\chi(G_{n,p})\leq \chi_g(G_{n,p})\leq c_2\chi(G_{n,p})$ for some absolute constants $1<c_1< c_2$. We also prove that if $G_{n,3}$ denotes a random $n$-vertex cubic graph, then w.h.p. $\chi_g(G_{n,3})=4$.

It is proved that community structure has great impact on epidemic spreading of BA networks which are highly sensitive to the targeted immunization of a small fraction of the most connected nodes. To find effective control strategies to fight the epidemic propagation in homogeneous networks with community structure, a model of pseudo-random network is presented with adjustable community structure and constant average degree. A novel immunization strategy is proposed which aims at immunizing the nodes with highest betweenness. Then the epidemic propagation with different spreading rate and different immunization fraction is investigated in this model. Simulations show that the new strategy can reduce the epidemic propagation velocity especially in the higher modularity networks. And the more the nodes are immunized the more efficient our strategy is.

A projective network model is a model that enables predictions to be made based on a subsample of the network data, with the predictions remaining unchanged if a larger sample is taken into consideration. An exchangeable model is a model that does not depend on the order in which nodes are sampled. Despite a large variety of non-equilibrium (growing) and equilibrium (static) sparse complex network models that are widely used in network science, how to reconcile sparseness (constant average degree) with the desired statistical properties of projectivity and exchangeability is currently an outstanding scientific problem. Here we propose a network process with hidden variables which is projective and can generate sparse power-law networks. Despite the model not being exchangeable, it can be closely related to exchangeable uncorrelated networks as indicated by its information theory characterization and its network entropy. The use of the proposed network process as a null model is here tested on real data, indicating that the model offers a promising avenue for statistical network modelling.

In his pioneering paper on matroids in 1935, Whitney obtained a characterization for binary matroids and left a comment at end of the paper that the problem of characterizing graphic matroids is the same as that of characterizing matroids which correspond to matrices (mod 2) with exactly two ones in each column. Later on Tutte obtained a characterization of graphic matroids in terms of forbidden minors in 1959. It is clear that Whitney indicated about incidence matrices of simple undirected graphs. Here we introduce the concept of a segment binary matroid which corresponds to matrices over $\mathbb{Z}_2$ which has the consecutive $1$'s property (i.e., $1$'s are consecutive) for columns and obtained a characterization of graphic matroids in terms of this. In fact, we introduce a new representation of simple undirected graphs in terms of some vectors of finite dimensional vector spaces over $\mathbb{Z}_2$ which satisfy consecutive $1$'s property. The set of such vectors is called a coding sequence of a graph $G$. Among all such coding sequences we identify the one which is unique for a class of isomorphic graphs. We call it the code of the graph. We characterize several classes of graphs in terms of coding sequences. It is shown that a graph $G$ with $n$ vertices is a tree if and only if any coding sequence of $G$ is a basis of the vector space $\mathbb{Z}_2^{n-1}$ over $\mathbb{Z}_2$. Moreover considering coding sequences as binary matroids, we obtain a characterization for simple graphic matroids and found a necessary and sufficient condition for graph isomorphism in terms of a special matroid isomorphism between their corresponding coding sequences. For this, we introduce the concept of strong isomorphisms of segment binary matroids and show that two simple (undirected) graphs are isomorphic if and only if their canonical sequences are strongly isomorphic segment binary matroids.

Let G be a given graph (modelling a communication network) which we assume suffers from static edge faults: That is we let each edge of G be present independently with probability p (or absent with fault probability f = 1 - p). In particular we are interested in robustness results for the case that the graph G itself is a random member of the class of all regular graphs with given degree. Our result is: If the degree d is fixed then p = 1/(d - 1) is a threshold probability for the existence of a linear-sized component in a faulty version of almost all random regular graphs. We show: If each edge of an arbitrary graph G with maximum degree bounded above by d is present with, probability p = Λ/(d - 1) where Λ 1 then for almost all G the faulted version of G has a linear-sized component with high probability. Note that these results imply some kind of optimality of random regular graphs among the class of graphs with the same degree bound. The theme is: Use the known expansion properties of almost all random regular graphs to obtain strong robustness results. This has not been done systematically before.

We investigate recursive relations for the Grothendieck classes of the affine graph hypersurface complements of melonic graphs. We compute these classes explicitly for several families of melonic graphs, focusing on the case of graphs with valence- 4 internal vertices, relevant to CTKT tensor models. The results hint at a complex and interesting structure in terms of divisibility relations or nontrivial relations between classes of graphs in different families. Using the recursive relations, we prove that the Grothendieck classes of all melonic graphs are positive as polynomials in the class of the moduli space \mathcal M_{0,4} . We also conjecture that the corresponding polynomials are log-concave , on the basis of hundreds of explicit computations.

A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for M AXIMUM C LIQUE on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics ’90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show that the disjoint union of two odd cycles is never the complement of a disk graph nor of a unit (3-dimensional) ball graph. From that fact and existing results, we derive a simple QPTAS and a subexponential algorithm running in time 2 Õ( n 2/3 ) for M AXIMUM C LIQUE on disk and unit ball graphs. We then obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VC-dimension, and linear independence number. This, in combination with our structural results, yields a randomized EPTAS for M AX C LIQUE on disk and unit ball graphs. M AX C LIQUE on unit ball graphs is equivalent to finding, given a collection of points in R 3 , a maximum subset of points with diameter at most some fixed value. In stark contrast, M AXIMUM C LIQUE on ball graphs and unit 4-dimensional ball graphs, as well as intersection graphs of filled ellipses (even close to unit disks) or filled triangles is unlikely to have such algorithms. Indeed, we show that, for all those problems, there is a constant ratio of approximation that cannot be attained even in time 2 n 1−ɛ , unless the Exponential Time Hypothesis fails.

In this paper, we consider the minimum maximal matching problem in some classes of graphs such as regular graphs. We show that the minimum maximal matching problem is NP-hard even in regular bipartite graphs, and a polynomial time exact algorithm is given for almost complete regular bipartite graphs. From the approximation point of view, it is well known that any maximal matching guarantees the approximation ratio of 2 but surprisingly very few improvements have been obtained. In this paper we give improved approximation ratios for several classes of graphs. For example any algorithm is shown to guarantee an approximation ratio of (2-o(1)) in graphs with high average degree. We also propose an algorithm guaranteeing for any graph of maximum degree Δ an approximation ratio of (2−1/Δ), which slightly improves the best known results. In addition, we analyse a natural linear-time greedy algorithm guaranteeing a ratio of (2−23/18k) in k-regular graphs admitting a perfect matching.

Bipartite bithreshold graphs

In this paper, we study the class of graphs Gm,n that have the same degree sequence as two disjoint cliques Km and Kn, as well as the class G¯m,n of the complements of such graphs. The problems of finding a maximum clique and a maximum independent set are NP-hard on Gm,n. Therefore, looking for upper and lower bounds for the clique and independence numbers of such graphs is a challenging task. In this article, we obtain such bounds, as well as other related results. In particular, we consider the class of regular graphs, which are degree-equivalent to arbitrarily many identical cliques, as well as such graphs of bounded degree.

We study the kernelization complexity of structural parameterizations of the Vertex Cover problem. Here, the goal is to find a polynomial-time preprocessing algorithm that can reduce any instance $(G,k)$ of the Vertex Cover problem to an equivalent one, whose size is polynomial in the size of a predetermined complexity parameter of $G$. A long line of previous research deals with parameterizations based on the number of vertex deletions needed to reduce $G$ to a member of a simple graph class $\mathcal{F}$, such as forests, graphs of bounded tree-depth, and graphs of maximum degree two. We set out to find the most general graph classes $\mathcal{F}$ for which Vertex Cover parameterized by the vertex-deletion distance of the input graph to $\mathcal{F}$ admits a polynomial kernelization. We give a complete characterization of the minor-closed graph families $\mathcal{F}$ for which such a kernelization exists. We introduce a new graph parameter called bridge-depth, and prove that a polynomial kernelization exists if and only if $\mathcal{F}$ has bounded bridge-depth. The proof is based on an interesting connection between bridge-depth and the size of minimal blocking sets in graphs, which are vertex sets whose removal decreases the independence number.

Accordion graphs: Hamiltonicity, matchings and isomorphism with quartic circulants

This chapter describes the history of graphs and introduces the main definitions of graphs. It discusses some particular classes of graphs that we can meet in philosophy and provides some examples of them. The chapter presents a list of well-known graphs that we can put together according to their name. A graph generally has different kinds of drawings, but all of these are isomorphic. So, a graph is in fact a class of graphs generally defined up to an isomorphism. Even when graphs are non-separable, they always admit subgraphs. Informally speaking, the reconstruction conjecture says that graphs are determined uniquely by their subgraphs. The conjecture has been verified for a number of infinite classes of graphs: regular graphs, trees, disconnected graphs, unit interval graphs, separable graphs without end vertices, maximal planar graphs, maximal outerplanar graphs, outerplanar graphs and critical blocks.

A thrackle is a drawing of a graph in which any two vertex-disjoint edges cross exactly once and incident edges do not cross. A graph that has a thrackle drawing is thracklable. In the past three decades, mathematicians investigated a number of variations of thrackles by relaxing or changing the required number of crossings on edges or restricting the placement of vertices on the surface (e.g., restricting to outerdrawings, in which vertices are restricted to a boundary curve of the surface). A graph is superthracklable if it can be drawn so that any two edges cross exactly once. We provide a simple proof for the following: a graph can be drawn on the plane so that every two edges cross an odd number of times if and only if it is superthracklable on the plane. A graph is outersuperthracklable if it has a drawing on a disc in which every vertex is on the boundary of the disc and every pair of edges cross exactly once. We introduce some natural generalisations of outersuperthracklable graphs and we show that these classes of graphs are equivalent. Lastly, using the Hanani-Tutte characterisation of planar graphs we show that for any surface $\Sigma$, there is a relationship between the class of graphs that are not embeddable on $\Sigma$ and the class of graphs that are not superthracklable with respect to $\Sigma$. More specifically, we show how to construct, from any forbidden minor $G$ for embeddability in $\Sigma$, two infinite families of graphs that are not superthracklable with respect to $\Sigma$. This sheds further light on the relationship between some of Archdeacon and Stor's forbidden configurations for superthrackles and Kuratowski's forbidden minors for planarity.

The metric representation of a vertex v of a graph G is a finite vector representing distances of v with respect to vertices of some ordered subset S⊆V (G). If no suitable subset of S provides separate representations for each vertex of V(G), then the set S is referred to as a minimal resolving set. The metric dimension of G is the cardinality of the smallest (with respect to its cardinality) minimal resolving set. A resolving set S is secure if for any v∈V–S, there exists x∈S such that (S–{x})∪{v} is a resolving set. For various classes of graphs, the value of the secure resolving number is determined and defined. The secure metric dimension of the graph classes is being studied in this work. The results show that different graph families have different metric dimensions.