Size Of Subgraph Research Articles

Dependence Polynomials of some Graph Operations

Suppose G is a simple graph and ck = ck(G) denotes the number of complete subgraphs of size k in G. Then the dependence polynomial of G is defined as fG(x) = 1−c1x+c2x2−c3x3+⋯+(−1)nxn, where n is the size of the largest complete subgraph in G. In this paper, exact formulas for dependence polynomial of some graph operations are presented.

Distributed discovery of frequent subgraphs of a network using MapReduce

Discovery of frequent subgraphs of a network is a challenging and time-consuming process. Several heuristics and improvements have been proposed before. However, when the size of subgraphs or the s...

Fast Parallel All-Subgraph Enumeration Using Multicore Machines

Enumerating all subgraphs of an input graph is an important task for analyzing complex networks. Valuable information can be extracted about the characteristics of the input graph using all-subgraph enumeration. Notwithstanding, the number of subgraphs grows exponentially with growth of the input graph or by increasing the size of the subgraphs to be enumerated. Hence, all-subgraph enumeration is very time consuming when the size of the subgraphs or the input graph is big. We propose a parallel solution namedSubenumwhich in contrast to available solutions can perform much faster. Subenum enumerates subgraphs using edges instead of vertices, and this approach leads to a parallel and load-balanced enumeration algorithm that can have efficient execution on current multicore and multiprocessor machines. Also, Subenum uses a fast heuristic which can effectively accelerate non-isomorphism subgraph enumeration. Subenum can efficiently use external memory, and unlike other subgraph enumeration methods, it is not associated with the main memory limits of the used machine. Hence, Subenum can handle large input graphs and subgraph sizes that other solutions cannot handle. Several experiments are done using real-world input graphs. Compared to the available solutions, Subenum can enumerate subgraphs several orders of magnitude faster and the experimental results show that the performance of Subenum scales almost linearly by using additional processor cores.

On Schur’s conjecture in ℝ4

A diameter graph in ℝd is a graph in which vertices are points of a finite subset of ℝd and two vertices are joined by an edge if the distance between them is equal to the diameter of the vertex set. This paper is devoted to Schur’s conjecture, which asserts that any diameter graph on n vertices in ℝd contains at most n complete subgraphs of size d. It is known that Schur’s conjecture is true in dimensions d ≤ 3. We prove this conjecture for d = 4 and give a simple proof for d = 3.

PTAS for Densest $$k$$ k -Subgraph in Interval Graphs

Given an interval graph and integer $$k$$k, we consider the problem of finding a subgraph of size $$k$$k with a maximum number of induced edges, called densest k -subgraph problem in interval graphs. This problem is NP-hard even for chordal graphs (Perl and Corneil in Discret Appl Math 9(1):27---39, 1984), and there is probably no PTAS for general graphs (Khot and Subhash in SIAM J Comput 36(4):1025---1071, 2006). However, the exact complexity status for interval graphs is a long-standing open problem (Perl and Corneil in Discret Appl Math 9(1):27---39, 1984), and the best known approximation result is a $$ 3 $$3-approximation algorithm (Liazi et al. in Inf Process Lett 108(1):29---32, 2008). We shed light on the approximation complexity of finding a densest $$ k $$k-subgraph in interval graphs by presenting a polynomial-time approximation scheme (PTAS), that is, we show that there is an $$ (1+\epsilon ) $$(1+∈)-approximation algorithm for any $$ \epsilon > 0 $$∈>0, which is the first such approximation scheme for the densest $$ k $$k-subgraph problem in an important graph class without any further restrictions.

Acc-Motif: Accelerated Network Motif Detection.

Network motif algorithms have been a topic of research mainly after the 2002-seminal paper from Milo et al. [1], which provided motifs as a way to uncover the basic building blocks of most networks. Motifs have been mainly applied in Bioinformatics, regarding gene regulation networks. Motif detection is based on induced subgraph counting. This paper proposes an algorithm to count subgraphs of size k + 2 based on the set of induced subgraphs of size k. The general technique was applied to detect 3, 4 and 5-sized motifs in directed graphs. Such algorithms have time complexity O(a(G)m), O(m(2)) and O(nm(2)), respectively, where a(G) is the arboricity of G(V, E). The computational experiments in public data sets show that the proposed technique was one order of magnitude faster than Kavosh and FANMOD. When compared to NetMODE, acc-Motif had a slightly improved performance.

Beyond the Vizing's Bound for at Most Seven Colors

Let $G=(V,E)$ be a simple graph of maximum degree $\Delta$. The edges of $G$ can be colored with at most $\Delta +1$ colors by Vizing's theorem. We study lower bounds on the size of subgraphs of $G$ that can be colored with $\Delta$ colors. Vizing's theorem gives a bound of $\frac{\Delta}{\Delta+1}|E|$. This is known to be tight for cliques $K_{\Delta+1}$ when $\Delta$ is even. However, for $\Delta=3$ it was improved to $\frac{26}{31}|E|$ by Albertson and Haas [Discrete Math., 148 (1996), pp. 1--7] and later to $\frac{6}7|E|$ by Rizzi [Discrete Math., 309 (2009), pp. 4166--4170]. It is tight for $B_3$, the graph isomorphic to a $K_4$ with one edge subdivided. We improve previously known bounds for $\Delta\in\{3,\ldots,7\}$, under the assumption that for $\Delta=3,4,6$, graph $G$ is not isomorphic to $B_3$, $K_5$, and $K_7$, respectively. For $\Delta \geq 4$ these are the first results which improve over the Vizing's bound. We also show a new bound for subcubic multigraphs not isomorphic to $K_3$ with one ...

Explicit and probabilistic constructions of distance graphs with small clique numbers and large chromatic numbers

We study distance graphs with exponentially large chromatic numbers and without -cliques, that is, complete subgraphs of size . Explicit constructions of such graphs use vectors in the integer lattice. For a large class of graphs we find a sharp threshold for containing a -clique. This enables us to improve the lower bounds for the maximum of the chromatic numbers of such graphs. We give a new probabilistic approach to the construction of distance graphs without -cliques, and this yields better lower bounds for the maximum of the chromatic numbers for large .

Preventing Unraveling in Social Networks Gets Harder

The behavior of users in social networks is often observed to be affected by the actions of their friends. Bhawalkar et al. (ICALP '12) introduced a formal mathematical model for user engagement in social networks where each individual derives a benefit proportional to the number of its friends which are engaged. Given a threshold degree k the equilibrium for this model is a maximal subgraph whose minimum degree is at least k. However the dropping out of individuals with degrees less than k might lead to a cascading effect of iterated withdrawals such that the size of equilibrium subgraph becomes very small. To overcome this some special vertices called "anchors" are introduced: these vertices need not have large degree. Bhawalkar et al. considered the Anchored k-Core problem: Given a graph G and integers b, k and p do there exist sets of vertices B, H such that B is a subset of H, size of B is at most b and size of H is at least p, and every vertex v which is in H but not in B has degree at least k in the induced subgraph G[H]. They showed that the problem is NP-hard for all k greater equal 2, and gave some inapproximability and fixed-parameter intractability results. In this paper we give improved hardness results for this problem. In particular we show that the Anchored k-Core problem is W[1]-hard parameterized by p, even for k=3. This improves the result of Bhawalkar et al. (who show W[2]-hardness parameterized by b) as our parameter is always bigger since p is greater equal than b. Then we answer a question of Bhawalkar et al. by showing that the Anchored k-Core problem remains NP-hard on planar graphs for all k greater equal 3, even if the maximum degree of the graph is k+2. Finally we show that the problem is FPT on planar graphs parameterized by b for all k greater equal 7.

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.

Steiner tree methods for optimal sub-network identification: an empirical study

BackgroundAnalysis and interpretation of biological networks is one of the primary goals of systems biology. In this context identification of sub-networks connecting sets of seed proteins or seed genes plays a crucial role. Given that no natural node and edge weighting scheme is available retrieval of a minimum size sub-graph leads to the classical Steiner tree problem, which is known to be NP-complete. Many approximate solutions have been published and theoretically analyzed in the computer science literature, but far less is known about their practical performance in the bioinformatics field.ResultsHere we conducted a systematic simulation study of four different approximate and one exact algorithms on a large human protein-protein interaction network with ~14,000 nodes and ~400,000 edges. Moreover, we devised an own algorithm to retrieve a sub-graph of merged Steiner trees. The application of our algorithms was demonstrated for two breast cancer signatures and a sub-network playing a role in male pattern baldness.ConclusionWe found a modified version of the shortest paths based approximation algorithm by Takahashi and Matsuyama to lead to accurate solutions, while at the same time being several orders of magnitude faster than the exact approach. Our devised algorithm for merged Steiner trees, which is a further development of the Takahashi and Matsuyama algorithm, proved to be useful for small seed lists. All our implemented methods are available in the R-package SteinerNet on CRAN (http://www.r-project.org) and as a supplement to this paper.

Data Structures and Algorithms for Counting Problems on Graphs using GPU

The availability and utility of large numbers of Graphical Processing Units (GPUs) have enabled parallel computations using extensive multi-threading. Sequential access to global memory and contention at the size-limited shared memory have been main impediments to fully exploiting potential performance in architectures having a massive number of GPUs. After performing extensive study of data structures and complexity analysis of various data access methodologies, we propose novel memory storage and retrieval techniques that enable parallel graph computations to overcome the above issues. More specifically, given a graph G = (V,E) and an integer k <= |V |, we provide both storage techniques and algorithms to count the number of: a) connected subgraphs of size k; b) k cliques; and c) k independent sets, all of which can be exponential in number. Our storage techniques are based on creating a breadth-first search tree and storing it along with non-tree edges in a novel way. Our experiments solve the above mentioned problems by using both naive and advanced data structures on the CPU and GPU. Speedup is achieved by solving the problems on the GPU even using a brute-force approach as compared to the implementations on the CPU. Utilizing the knowledge of BFS-tree properties, the performance gain on the GPU increases and ultimately outperforms the CPU by a factor of at least 5 for graphs that completely fit in the shared memory and by a factor of 10 for larger graphs stored using the global memory. The counting problems mentioned above have many uses, including the analysis of social networks.

Arboricity, [formula omitted]-index, and dynamic algorithms

We propose a new data structure for manipulating graphs, called h-graph, which is particularly suited for designing dynamic algorithms. The structure itself is simple, consisting basically of a triple of elements, for each vertex of the graph. The overall size of all triples is O(n+m), for a graph with n vertices and m edges. We describe algorithms for performing the basic operations related to dynamic applications, as insertions and deletions of vertices or edges, and adjacency queries. The data structure employs a technique first described by Chiba and Nishizeki [Chiba, Nishizeki, Arboricity and subgraph listing algorithms, SIAM J. Comput. 14 (1) (1985) 210–223], and relies on the arboricity of graphs. Using the proposed data structure, we describe several dynamic algorithms for solving problems as listing the cliques of a given size, recognizing diamond-free graphs, and finding simple, simplicial and dominated vertices. These algorithms are the first of their kind to be proposed in the literature. In fact, the dynamic algorithms for the above problems lead directly to new static algorithms, and using the data structure we also design new static algorithms for the problems of counting subgraphs of size 4, recognizing cop-win graphs and recognizing strongly chordal graphs. The complexities of all of the proposed static algorithms improve over the complexities of the so far existing algorithms, for graphs of low arboricity. In addition, for the problems of counting subgraphs of size 4 and recognizing diamond-free graphs, the improvement is general.

Extended dynamic subgraph statistics using [formula omitted]-index parameterized data structures

We present techniques for maintaining subgraph frequencies in a dynamic graph, using data structures that are parameterized in terms of h, the h-index of the graph. Our methods extend previous results of Eppstein and Spiro for maintaining statistics for undirected subgraphs of size three to directed subgraphs and to subgraphs of size four. For the directed case, we provide a data structure to maintain counts for all 3-vertex induced subgraphs in O(h) amortized time per update. For the undirected case, we maintain the counts of size-four subgraphs in O(h2) amortized time per update. These extensions enable a number of new applications in Bioinformatics and Social Networking research.

Progressive Frequency Offset Compensation in Turbo Receivers

Based on the recently-proposed iterative receiver by Colavolpe et al., we extend it to further incorporate the mechanism of frequency offset compensation. As a progressive approach, the factor graph of a long frame is first divided into numerous subgraphs of short block, iterative decoding can successfully start its work as the effect of frequency offset can be well mitigated due to the small size of subgraph. As the iteration goes on, the extrinsic information from decoder can be employed to get finer estimate of the frequency offset. The system performance can be steadily improved as the block size is progressively expanded to the final frame size.

On the Clique Numbers of Non-commuting Graphs of Certain Groups

Let G be a non-abelian group. The non-commuting graph [Formula: see text] of G is defined as the graph whose vertex set is the non-central elements of G and two vertices are joint if and only if they do not commute. In a finite simple graph Γ, the maximum size of complete subgraphs of Γ is called the clique number of Γ and denoted by ω(Γ). In this paper, we characterize all non-solvable groups G with [Formula: see text], where 57 is the clique number of the non-commuting graph of the projective special linear group PSL (2,7). We also determine [Formula: see text] for all finite minimal simple groups G.

Connecting face hitting sets in planar graphs

We show that any face hitting set of size n of a connected planar graph with a minimum degree of at least 3 is contained in a connected subgraph of size 5 n − 6 . Furthermore we show that this bound is tight by providing a lower bound in the form of a family of graphs. This improves the previously known upper and lower bound of 11 n − 18 and 3 n respectively by Grigoriev and Sitters. Our proof is valid for simple graphs with loops and generalizes to graphs embedded in surfaces of arbitrary genus.

Bipartite Divisor Graphs for Integer Subsets

Inspired by connections described in a recent paper by Mark L. Lewis, between the common divisor graph Γ(X) and the prime vertex graph Δ(X), for a set X of positive integers, we define the bipartite divisor graph B(X), and show that many of these connections flow naturally from properties of B(X). In particular we establish links between parameters of these three graphs, such as number and diameter of components, and we characterise bipartite graphs that can arise as B(X) for some X. Also we obtain necessary and sufficient conditions, in terms of subconfigurations of B(X), for one of Γ(X) or Δ(X) to contain a complete subgraph of size 3 or 4.

On even-cycle-free subgraphs of the hypercube

Abstract It is shown that the size of a subgraph of the hypercube Q n without a cycle of length 4 k + 2 for integer k ⩾ 3 is of order o ( | E ( Q n ) | ) .

Approximating the Number of Network Motifs

The World Wide Web, the Internet, coupled biological and chemical systems, neural networks, and social interacting species are only a few examples of systems comprising a large number of highly interconnected dynamical units. These networks contain characteristic patterns, _network motifs_, that occur far more often than in randomized networks with the same degree sequence. Several algorithms have been suggested for counting or detecting the number of occurrences of network motifs as trees and bounded treewidth subgraphs of size _O_(log _n_), at most 7 for some motifs. In addition, _local motif counting_, counting the number of motifs in which a node participates, was recently suggested as a method of classifying nodes in the network. The premise is that the distribution of motifs in which a node participates is an indication of its function in the network. Therefore, local counting of network motifs provides a major challenge. However, no such practical algorithm exists other than local counting of triangles. We present several algorithms with time complexity _O_(((3_e_)^<em>k</em> · _n_ · |_E_| · log <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="uinm_a_10390645_o_uf0001.gif"></inline-graphic>)/∊²) that approximate for every vertex the number of occurrences of the motif in which the vertex participates, for _k_-length cycles and _k_-length cycles with a chord, where _k_ = _O_(log _n_), and algorithms with time complexity _O_((_n_ · |_E_| · log <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="uinm_a_10390645_o_uf0001.gif"></inline-graphic>)/∊² + |_E_|² · log _n_ + |_E_| · _n_ log _n_) that approximate for every vertex the number of noninduced occurrences of the motif in which the vertex participates for all motifs of size four. In addition, we show algorithms that approximate the total number of occurrences of these network motifs when no efficient algorithm exists. Some of our algorithms use the “color-coding” technique.