Largest Subgraph Research Articles

English

We consider, for a positive integer $k$, induced subgraphs in which each component has order at most $k$. Such a subgraph is said to be $k$-divided. We show that finding large induced subgraphs with this property is NP-complete. We also consider a related graph-coloring problem: how many colors are required in a vertex coloring in which each color class induces a $k$-divided subgraph. We show that the problem of determining whether some given number of colors suffice is NP-complete, even for $2$-coloring a planar triangle-free graph. Lastly, we consider Ramsey-type problems where graphs or their complements with large enough order must contain a large $k$-divided subgraph. We study the asymptotic behavior of ``$k$-divided Ramsey numbers''. We conclude by mentioning a number of open problems.

Refining the clustering coefficient for analysis of social and neural network data

In this paper we show how a deeper analysis of the clustering coefficient in a network can be used to assess functional connections in the human brain. Our metric of edge clustering centrality considers the frequency at which an edge appears across all local subgraphs that are induced by each vertex and its neighbors. This analysis is tied to a problem from structural graph theory in which we seek the largest subgraph that is a Cartesian product of two complete bipartite graphs \(K_{1,m}\) and \(K_{1,1}\). We investigate this property and compare it to other known edge centrality metrics. Finally, we apply the property of clustering centrality to an analysis of functional MRI data obtained, while healthy participants pantomimed object use or identified objects.

MTP: discovering high quality partitions in real world graphs

Given a real world graph, how can we find a large subgraph whose partition quality is much better than the original? How can we use a partition of that subgraph to discover a high quality global partition? Although graph partitioning especially with balanced sizes has received attentions in various applications, it is known NP-hard, and also known that there is no good cut at a large scale for real graphs. In this paper, we propose a novel approach for graph partitioning. Our first focus is on finding a large subgraph with high quality partitions, in terms of conductance. Despite the difficulty of the task for the whole graph, we observe that there is a large connected subgraph whose partition quality is much better than the original. Our proposed method MTP finds such a subgraph by removing "hub" nodes with large degrees, and taking the remaining giant connected component. Further, we extend MTP to gb MTP (Global Balanced MTP) for discovering a global balanced partition. gb MTP attaches the excluded nodes in MTP to the partition found by MTP in a greedy way. In experiments, we demonstrate that MTP finds a subgraph of a large size with low conductance graph partitions, compared with competing methods. We also show that the competitors cannot find connected subgraphs while our method does, by construction. This improvement in partition quality for the subgraph is especially noticeable for large scale cuts--for a balanced partition, down to 14 % of the original conductance with the subgraph size 70 % of the total. As a result, the found subgraph has clear partitions at almost all scales compared with the original. Moreover, gb MTP generally discovers global balanced partitions whose conductance are lower than those found by METIS, the state-of-the-art graph partitioning method.

On Large Subgraphs with Small Chromatic Numbers Contained in Distance Graphs

It is proved that each distance graph on a plane has an induced subgraph with a chromatic number that is at most 4 containing over 91% of the vertices of the original graph. This result is used to obtain the asymptotic growth rate for a threshold probability that a random graph is isomorphic to a certain distance graph on a plane. Several generalizations to larger dimensions are proposed.

Multivariate Algorithmics for Finding Cohesive Subnetworks

Community detection is an important task in the analysis of biological, social or technical networks. We survey different models of cohesive graphs, commonly referred to as clique relaxations, that are used in the detection of network communities. For each clique relaxation, we give an overview of basic model properties and of the complexity of the problem of finding large cohesive subgraphs under this model. Since this problem is usually NP-hard, we focus on combinatorial fixed-parameter algorithms exploiting typical structural properties of input networks.

A random graph model of density thresholds in swarming cells

Swarming behaviour is a type of bacterial motility that has been found to be dependent on reaching a local density threshold of cells. With this in mind, the process through which cell-to-cell interactions develop and how an assembly of cells reaches collective motility becomes increasingly important to understand. Additionally, populations of cells and organisms have been modelled through graphs to draw insightful conclusions about population dynamics on a spatial level. In the present study, we make use of analogous random graph structures to model the formation of large chain subgraphs, representing interactions between multiple cells, as a random graph Markov process. Using numerical simulations and analytical results on how quickly paths of certain lengths are reached in a random graph process, metrics for intercellular interaction dynamics at the swarm layer that may be experimentally evaluated are proposed.

Finding large degree-anonymous subgraphs is hard

A graph is said to be k-anonymous for an integer k, if for every vertex in the graph there are at least k−1 other vertices with the same degree. We examine the computational complexity of making a given undirected graph k-anonymous either through at most s vertex deletions or through at most s edge deletions; the corresponding problem variants are denoted by Anonym V-Del and Anonym E-Del.We present a variety of hardness results, most of them hold for both problems. The two variants are intractable from the parameterized as well as from the approximation point of view. In particular, we show that both variants remain NP-hard on very restricted graph classes like trees even if k=2. We further prove that both variants are W[1]-hard with respect to the combined parameter solutions size s and anonymity level k. With respect to approximability, we obtain hardness results showing that neither variant can be approximated in polynomial time within a factor better than n12 (unless P=NP). Furthermore, for the optimization variants where the solution size s is given and the task is to maximize the anonymity level k, this inapproximability result even holds if we allow a running time of f(s)⋅nO(1) for any computable function f. On the positive side, we classify both problem variants as fixed-parameter tractable with respect to the combined parameter solution size s and maximum degree Δ.

The complexity of proving that a graph is Ramsey

We say that a graph with n vertices is c-Ramsey if it does not contain either a clique or an independent set of size c log n. We define a CNF formula which expresses this property for a graph G. We show a superpolynomial lower bound on the length of resolution proofs that G is c-Ramsey, for every graph G. Our proof makes use of the fact that every c-Ramsey graph must contain a large subgraph with some properties typical for random graphs.

Fractional Turan's theorem and bounds for the chromatic number

Turan's theorem gives conditions for finding large complete subgraphs in a graph in terms of the number of edges. In this work we look for families of graphs in which there is a similar theorem, but which will allow us to find larger complete subgraphs. This question is motivated by a geometric result by Katchalski and Liu concerning a fractional Helly theorem.We present results in two directions. On the one hand, we characterize the families of graphs for which there exists a constant β such that a proportion of α edges guarantees the existence of a complete subgraph using a proportion of αβ of the total number of vertices. On the other hand, we give a criterion to guarantee that in a family of graphs any fixed proportion of the edges is sufficient to conclude that the order of the largest complete subgraph goes to infinity as the number of vertices goes to infinity. These results are obtained by giving an upper bound for the chromatic number of a graph in terms of the clique number and an arbitrarily small proportion of the number of vertices.

How does the core sit inside the mantle?

The k-core, defined as the largest subgraph of minimum degree k, of the random graph G(n,p) has been studied extensively. In a landmark paper Pittel, Wormald and Spencer [Journal of Combinatorial Theory, Series B 67 (1996) 111–151] determined the threshold dk for the appearance of an extensive k-core. Here we derive a multi-type Galton-Watson branching process that describes precisely how the k-core is “embedded” into the random graph for any k≥3 and any fixed average degree d=np>dk. This generalises prior results on, e.g., the internal structure of the k-core.

Transitive graphs uniquely determined by their local structure

We show that the "grandfather graph" has the following property: it is the unique completion to a transitive graph of a large enough finite subgraph of itself.

Large Subgraphs without Short Cycles

We study two extremal problems about subgraphs excluding a family $\F$ of graphs. i) Among all graphs with $m$ edges, what is the smallest size $f(m,\F)$ of a largest $\F$--free subgraph? ii) Among all graphs with minimum degree $\delta$ and maximum degree $\Delta$, what is the smallest minimum degree $h(\delta,\Delta,\F)$ of a spanning $\F$--free subgraph with largest minimum degree? These questions are easy to answer for families not containing any bipartite graph. We study the case where $\F$ is composed of all even cycles of length at most $2r$, $r\geq 2$. In this case, we give bounds on $f(m,\F)$ and $h(\delta,\Delta,\F)$ that are essentially asymptotically tight up to a logarithmic factor. In particular for every graph $G$, we show the existence of subgraphs with arbitrarily high girth, and with either many edges or large minimum degree. These subgraphs are created using probabilistic embeddings of a graph into extremal graphs.

K-core percolation on multiplex networks.

We generalize the theory of k-core percolation on complex networks to k-core percolation on multiplex networks, where k≡(k(1),k(2),...,k(M)). Multiplex networks can be defined as networks with vertices of one kind but M different types of edges, representing different types of interactions. For such networks, the k-core is defined as the largest subgraph in which each vertex has at least k(i) edges of each type, i=1,2,...,M. We derive self-consistency equations to obtain the birth points of the k-cores and their relative sizes for uncorrelated multiplex networks with an arbitrary degree distribution. To clarify our general results, we consider in detail multiplex networks with edges of two types and solve the equations in the particular case of Erdős-Rényi and scale-free multiplex networks. We find hybrid phase transitions at the emergence points of k-cores except the (1,1)-core for which the transition is continuous. We apply the k-core decomposition algorithm to air-transportation multiplex networks, composed of two layers, and obtain the size of (k(1),k(2))-cores.

Detecting community structure via the maximal sub-graphs and belonging degrees in complex networks

Community structure is a common phenomenon in complex networks, and it has been shown that some communities in complex networks often overlap each other. So in this paper we propose a new algorithm to detect overlapping community structure in complex networks. To identify the overlapping community structure, our algorithm firstly extracts fully connected sub-graphs which are maximal sub-graphs from original networks. Then two maximal sub-graphs having the key pair-vertices can be merged into a new larger sub-graph using some belonging degree functions. Furthermore we extend the modularity function to evaluate the proposed algorithm. In addition, overlapping nodes between communities are founded successfully. Finally we report the comparison between the modularity and the computational complexity of the proposed algorithm with some other existing algorithms. The experimental results show that the proposed algorithm gives satisfactory results.

Uncovering the overlapping community structure of complex networks by maximal cliques

In this paper, a unique algorithm is proposed to detect overlapping communities in the un-weighted and weighted networks with considerable accuracy. The maximal cliques, overlapping vertex, bridge vertex and isolated vertex are introduced. First, all the maximal cliques are extracted by the algorithm based on the deep and bread searching. Then two maximal cliques can be merged into a larger sub-graph by some given rules. In addition, the proposed algorithm successfully finds overlapping vertices and bridge vertices between communities. Experimental results using some real-world networks data show that the performance of the proposed algorithm is satisfactory.

Approximately Counting Embeddings into Random Graphs

LetHbe a graph, and letCH(G) be the number of (subgraph isomorphic) copies ofHcontained in a graphG. We investigate the fundamental problem of estimatingCH(G). Previous results cover only a few specific instances of this general problem, for example the case whenHhas degree at most one (the monomer-dimer problem). In this paper we present the first general subcase of the subgraph isomorphism counting problem, which is almost always efficiently approximable. The results rely on a new graph decomposition technique. Informally, the decomposition is a labelling of the vertices such that every edge is between vertices with different labels, and for every vertex all neighbours with a higher label have identical labels. The labelling implicitly generates a sequence of bipartite graphs, which permits us to break the problem of counting embeddings of large subgraphs into that of counting embeddings of small subgraphs. Using this method, we present a simple randomized algorithm for the counting problem. For all decomposable graphsHand all graphsG, the algorithm is an unbiased estimator. Furthermore, for all graphsHhaving a decomposition where each of the bipartite graphs generated is small and almost all graphsG, the algorithm is a fully polynomial randomized approximation scheme.We show that the graph classes ofHfor which we obtain a fully polynomial randomized approximation scheme for almost allGincludes graphs of degree at most two, bounded-degree forests, bounded-width grid graphs, subdivision of bounded-degree graphs, and major subclasses of outerplanar graphs, series-parallel graphs and planar graphs of large girth, whereas unbounded-width grid graphs are excluded. Moreover, our general technique can easily be applied to proving many more similar results.

On large subgraphs of a distance graph which have small chromatic number

On large subgraphs of a distance graph which have small chromatic number

Dynamics of large cooperative pulsed-coupled networks

We study the deterministic dynamics of networks N composed by m non identical, mutually pulse-coupled cells. We assume weighted, asymmetric and positive (cooperative) interactions among the cells, and arbitrarily large values of m. We consider two cases of the network's graph: the complete graph, and the existence of a large core (i.e. a large complete subgraph). First, we prove that the system periodically eventually synchronizes with a natural "spiking period" \ p >=1, and that if the cells are mutually structurally identical or similar, then the synchronization is complete (p= 1) . Second, we prove that the amount of information H that N generates or processes equals log p. Therefore, if N completely synchronizes, the information is null. Finally, we prove that N protects the cells from their risk of death.

Automated generation of conjectures on forbidden subgraph characterization

Given a class of graphs F, a forbidden subgraph characterization (FSC) is a set of graphs H such that a graph G belongs to F if and only if no graph of H is isomorphic to an induced subgraph of G. FSCs play a key role in graph theory, and are at the center of many important results obtained in that field. In this paper, we present novel methods that automate the generation of conjectures on FSCs. Since most classes of graphs do not have such characterization, we also describe methods to find less restrictive results in the form of necessary or sufficient conditions to characterize a class of graphs with forbidden subgraphs. Furthermore, while these methods require to explore a possibly infinite search space, we present an enumerative technique that guarantees the discovery of characterizations involving forbidden subgraphs with a limited number of vertices. Another technique, which enables the discovery of characterizations with much larger subgraphs through the use of a heuristic search, is also described. In our experiments, we use these methods to find new theorems on the characterization of well-known graph classes.

Note on Upper Density of Quasi-Random Hypergraphs

In 1964, Erdős proved that for any $\alpha > 0$, an $l$-uniform hypergraph $G$ with $n \geq n_0(\alpha, l)$ vertices and $\alpha \binom{n}{l}$ edges contains a large complete $l$-equipartite subgraph. This implies that any sufficiently large $G$ with density $\alpha > 0$ contains a large subgraph with density at least $l!/l^l$.In this note we study a similar problem for $l$-uniform hypergraphs $Q$ with a weak quasi-random property (i.e. with edges uniformly distributed over the sufficiently large subsets of vertices). We prove that any sufficiently large quasi-random $l$-uniform hypergraph $Q$ with density $\alpha > 0$ contains a large subgraph with density at least $\frac{(l-1)!}{l^{l-1}-1}$. In particular, for $l=3$, any sufficiently large such $Q$ contains a large subgraph with density at least $\frac{1}{4}$ which is the best possible lower bound.We define jumps for quasi-random sequences of $l$-graphs and our result implies that every number between 0 and $\frac{(l-1)!}{l^{l-1}-1}$ is a jump for quasi-random $l$-graphs. For $l=3$ this interval can be improved based on a recent result of Glebov, Král' and Volec. We prove that every number between [0, 0.3192) is a jump for quasi-random $3$-graphs.