Cliques In Graphs Research Articles

Large Cliques in Sparse Random Intersection Graphs

An intersection graph defines an adjacency relation between subsets $S_1,\dots, S_n$ of a finite set $W=\{w_1,\dots, w_m\}$: the subsets $S_i$ and $S_j$ are adjacent if they intersect. Assuming that the subsets are drawn independently at random according to the probability distribution $\mathbb{P}(S_i=A)=P(|A|){\binom{m}{|A|}}^{-1}$, $A\subseteq W$, where $P$ is a probability on $\{0, 1, \dots, m\}$, we obtain the random intersection graph $G=G(n,m,P)$. We establish the asymptotic order of the clique number $\omega(G)$ of a sparse random intersection graph as $n,m\to+\infty$. For $m = \Theta(n)$ we show that the maximum clique is of size $(1-\alpha/2)^{-\alpha/2}n^{1-\alpha/2}(\ln n)^{-\alpha/2}(1+o_P(1))$ in the case where the asymptotic degree distribution of $G$ is a power-law with exponent $\alpha \in (1,2)$. It is of size $\frac {\ln n} {\ln \ln n}(1+o_P(1))$ if the degree distribution has bounded variance, i.e., $\alpha>2$. We construct a simple polynomial-time algorithm which finds a clique of the optimal order $\omega(G) (1-o_P(1))$.

Algorithms Parameterized by Vertex Cover and Modular Width, Through Potential Maximal Cliques

In this paper we give upper bounds on the number of minimal separators and potential maximal cliques of graphs w.r.t. two graph parameters, namely vertex cover ( $${\text {vc}}$$ ) and modular width ( $${\text {mw}}$$ ). We prove that for any graph, the number of its minimal separators is $${\mathcal {O}}^*(3^{{\text {vc}}})$$ and $${\mathcal {O}}^*(1.6181^{{\text {mw}}})$$ , and the number of potential maximal cliques is $${\mathcal {O}}^*(4^{{\text {vc}}})$$ and $${\mathcal {O}}^*(1.7347^{{\text {mw}}})$$ , and these objects can be listed within the same running times (The $${\mathcal {O}}^*$$ notation suppresses polynomial factors in the size of the input). Combined with known applications of potential maximal cliques, we deduce that a large family of problems, e.g., Treewidth, Minimum Fill-in, Longest Induced Path, Feedback vertex set and many others, can be solved in time $${\mathcal {O}}^*(4^{{\text {vc}}})$$ or $${\mathcal {O}}^*(1.7347^{{\text {mw}}})$$ . With slightly different techniques, we prove that the Treedepth problem can be also solved in single-exponential time, for both parameters.

Pushdown reachability with constant treewidth

We consider the problem of reachability in pushdown graphs. We study the problem for pushdown graphs with constant treewidth. Even for pushdown graphs with treewidth 1, for the reachability problem we establish the following: (i) the problem is PTIME-complete, and (ii) any subcubic algorithm for the problem would contradict the k-clique conjecture and imply faster combinatorial algorithms for cliques in graphs.

View-Based 3-D Model Retrieval: A Benchmark.

View-based 3-D model retrieval is one of the most important techniques in numerous applications of computer vision. While many methods have been proposed in recent years, to the best of our knowledge, there is no benchmark to evaluate the state-of-the-art methods. To tackle this problem, we systematically investigate and evaluate the related methods by: 1) proposing a clique graph-based method and 2) reimplementing six representative methods. Moreover, we concurrently evaluate both hand-crafted visual features and deep features on four popular datasets (NTU60, NTU216, PSB, and ETH) and one challenging real-world multiview model dataset (MV-RED) prepared by our group with various evaluation criteria to understand how these algorithms perform. By quantitatively analyzing the performances, we discover the graph matching-based method with deep features, especially the clique graph matching algorithm with convolutional neural networks features, can usually outperform the others. We further discuss the future research directions in this field.

Coloring the cliques of line graphs

The weak chromatic number, or clique chromatic number (CCHN) of a graph is the minimum number of colors in a vertex coloring, such that every maximal clique gets at least two colors. The weak chromatic index, or clique chromatic index (CCHI) of a graph is the CCHN of its line graph.Most of the results here are upper bounds for the CCHI, as functions of some other graph parameters, and contrasting with lower bounds in some cases. Algorithmic aspects are also discussed; the main result within this scope (and in the paper) shows that testing whether the CCHI of a graph equals 2 is NP-complete. We deal with the CCHN of the graph itself as well.

EXPLORATION OF STYLIZED FACTS IN THE ARTIFICIAL LIFE SYSTEM AVIDA

Population dynamics in the evolution, extinction, and re-evolution of various logic-function performing organisms is studied in the artificial life system, Avida. Following the work of Yedid (2009), we design an experiment involving two extinction regimes, pulse-extinction (corresponding to a random-kill event) and press-extinction (corresponding to a prolonged episode of rare resources). In addition, we study the effect of environmental topology (toroidal grid and clique graph). In the study of population dynamics, logarithmic returns are generally applied. The resulting distributions display a fat tail form of the power law: the more complex the logic function (in terms of NAND components), the broader the full width at half a maximum of the histogram. The power law exponents were in sound agreement with those of “real-life” populations and distributions. The distributions of evolutionary times, as well as post-extinction recovery periods, were very broad, and presumably had no standard deviations. Using 100 runs of 200,000 updates for each of the four cases (about 1 month of central processing unit time), we established the dynamics of the average population, with the effect of world topology.

Cliques in Graphs Excluding a Complete Graph Minor

This paper considers the following question: What is the maximum number of $k$-cliques in an $n$-vertex graph with no $K_t$-minor? This question generalises the extremal function for $K_t$-minors, which corresponds to the $k=2$ case. The exact answer is given for $t\leq 9$ and all values of $k$. We also determine the maximum total number of cliques in an $n$-vertex graph with no $K_t$-minor for $t\leq 9$. Several observations are made about the case of general $t$.

On clique convergence of graphs

Let G be a graph and KG be the set of all cliques of G, then the clique graph of G denoted by K(G) is the graph with vertex set KG and two elements Qi,Qj∈KG form an edge if and only if Qi∩Qj≠0̸. Iterated clique graphs are defined by K0(G)=G, and Kn(G)=K(Kn−1(G)) for n>0. In this paper we prove a necessary and sufficient condition for a clique graph K(G) to be complete when G=G1+G2, give a partial characterization for clique divergence of the join of graphs and prove that if G1, G2 are Clique-Helly graphs different from K1 and G=G1□G2, then K2(G)=G.

Determining majority in networks with local interactions and very small local memory

We study the problem of determining the majority type in an arbitrary connected network, each vertex of which has initially two possible types. The vertices may later change into other types, out of a set of a few additional possible types, and can interact in pairs only if they share an edge. Any (population) protocol is required to stabilize in the initial majority. First we prove that there does not exist any population protocol that always computes majority in any interaction graph by using at most 3 types per vertex. However this does not rule out the existence of a protocol with 3 types per vertex that is correct with high probability (whp). To this end, we examine an elegant and very natural majority protocol with 3 types per vertex, introduced in Angluin et al. (Distrib. Computing 21(2):87–102, 2008), whose performance has been analyzed for the clique graph. In particular, we study the performance of this protocol in arbitrary networks, under the probabilistic scheduler. We prove that, if the initial assignement of types to vertices is random, the protocol of Angluin et al. (Distrib. Computing 21(2):87–102, 2008) converges to the initial majority with probability higher than the probability of converging to the initial minority. In contrast, we show that the resistance of the protocol to failure when the underlying graph is a clique causes the failure of the protocol in general graphs.

EVALUATION OF THE USER STRATEGY ON 2D AND 3D CITY MAPS BASED ON NOVEL SCANPATH COMPARISON METHOD AND GRAPH VISUALIZATION

The paper is dealing with scanpath comparison of eye-tracking data recorded during case study focused on the evaluation of 2D and 3D city maps. The experiment contained screenshots from three map portals. Two types of maps were used - standard map and 3D visualization. Respondents’ task was to find particular point symbol on the map as fast as possible. <br><br> Scanpath comparison is one group of the eye-tracking data analyses methods used for revealing the strategy of the respondents. In cartographic studies, the most commonly used application for scanpath comparison is eyePatterns that output is hierarchical clustering and a tree graph representing the relationships between analysed sequences. During an analysis of the algorithm generating a tree graph, it was found that the outputs do not correspond to the reality. We proceeded to the creation of a new tool called ScanGraph. This tool uses visualization of cliques in simple graphs and is freely available at <a href="www.eyetracking.upol.cz/scangraph"target="_blank">www.eyetracking.upol.cz/scangraph</a>. <br><br> Results of the study proved the functionality of the tool and its suitability for analyses of different strategies of map readers. Based on the results of the tool, similar scanpaths were selected, and groups of respondents with similar strategies were identified. With this knowledge, it is possible to analyse the relationship between belonging to the group with similar strategy and data gathered from the questionnaire (age, sex, cartographic knowledge, etc.) or type of stimuli (2D, 3D map).

EVALUATION OF THE USER STRATEGY ON 2D AND 3D CITY MAPS BASED ON NOVEL SCANPATH COMPARISON METHOD AND GRAPH VISUALIZATION

Abstract. The paper is dealing with scanpath comparison of eye-tracking data recorded during case study focused on the evaluation of 2D and 3D city maps. The experiment contained screenshots from three map portals. Two types of maps were used - standard map and 3D visualization. Respondents’ task was to find particular point symbol on the map as fast as possible. Scanpath comparison is one group of the eye-tracking data analyses methods used for revealing the strategy of the respondents. In cartographic studies, the most commonly used application for scanpath comparison is eyePatterns that output is hierarchical clustering and a tree graph representing the relationships between analysed sequences. During an analysis of the algorithm generating a tree graph, it was found that the outputs do not correspond to the reality. We proceeded to the creation of a new tool called ScanGraph. This tool uses visualization of cliques in simple graphs and is freely available at www.eyetracking.upol.cz/scangraph. Results of the study proved the functionality of the tool and its suitability for analyses of different strategies of map readers. Based on the results of the tool, similar scanpaths were selected, and groups of respondents with similar strategies were identified. With this knowledge, it is possible to analyse the relationship between belonging to the group with similar strategy and data gathered from the questionnaire (age, sex, cartographic knowledge, etc.) or type of stimuli (2D, 3D map).

Simple and three-valued simple minimum coloring games

In this paper minimum coloring games are considered. We characterize the class of conflict graphs inducing simple or three-valued simple minimum coloring games. We provide an upper bound on the number of maximum cliques of conflict graphs inducing such games. Moreover, a characterization of the core is provided in terms of the underlying conflict graph. In particular, in case of a perfect conflict graph the core of an induced three-valued simple minimum coloring game equals the vital core.

SMT Solving for Functional Programming over Infinite Structures

We develop a simple functional programming language aimed at manipulating infinite, but first-order definable structures, such as the countably infinite clique graph or the set of all intervals with rational endpoints. Internally, such sets are represented by logical formulas that define them, and an external satisfiability modulo theories (SMT) solver is regularly run by the interpreter to check their basic properties. The language is implemented as a Haskell module.

An Exact Algorithm Based on MaxSAT Reasoning for the Maximum Weight Clique Problem

Recently, MaxSAT reasoning is shown very effective in computing a tight upper bound for a Maximum Clique (MC) of a (unweighted) graph. In this paper, we apply MaxSAT reasoning to compute a tight upper bound for a Maximum Weight Clique (MWC) of a wighted graph. We first study three usual encodings of MWC into weighted partial MaxSAT dealing with hard clauses, which must be satisfied in all solutions, and soft clauses, which are weighted and can be falsified. The drawbacks of these encodings motivate us to propose an encoding of MWC into a special weighted partial MaxSAT formalism, called LW (Literal-Weighted) encoding and dedicated for upper bounding an MWC, in which both soft clauses and literals in soft clauses are weighted. An optimal solution of the LW MaxSAT instance gives an upper bound for an MWC, instead of an optimal solution for MWC. We then introduce two notions called the Top-k literal failed clause and the Top-k empty clause to extend classical MaxSAT reasoning techniques, as well as two sound transformation rules to transform an LW MaxSAT instance. Successive transformations of an LW MaxSAT instance driven by MaxSAT reasoning give a tight upper bound for the encoded MWC. The approach is implemented in a branch-and-bound algorithm called MWCLQ. Experimental evaluations on the broadly used DIMACS benchmark, BHOSLIB benchmark, random graphs and the benchmark from the winner determination problem show that our approach allows MWCLQ to reduce the search space significantly and to solve MWC instances effectively. Consequently, MWCLQ outperforms state-of-the-art exact algorithms on the vast majority of instances. Moreover, it is surprisingly effective in solving hard and dense instances.

Detecting robust cliques in graphs subject to uncertain edge failures

This paper develops and compares several heuristic approaches, as well as an exact combinatorial branch-and-bound algorithm, for detecting maximum robust cliques in graphs subjected to multiple uncertain edge failures. The desired robustness properties are enforced using conditional value-at-risk measure. The proposed heuristics are adaptations of the well-known tabu search and GRASP methods, whereas the exact approach is an extension of Ostergard’s algorithm for the maximum clique problem. The results of computational experiments on DIMACS graph instances are reported.

ILP-assisted de novo drug design

De novo design of drugs uses the three-dimensional structure of a target protein (often called the receptor) to design molecules (or ligands) that could bind to the receptor and hence inhibit its functioning. Thus, unlike a ligand-based approach, this form of drug design does not require prior knowledge of inhibitors. In this paper, the three-dimensional structure of a receptor is used indirectly, in the form of molecular interaction fields of the receptor and small molecules (or probes). In addition, we also use domain-specific constraints encoding basic geometric and pharmacological requirements imposed by the target. Interaction energies of one or more targets with a set of probes are used to identify three-dimensional constraints that occur in many--preferably all--targets. In a graph-theoretic sense, the constraints are (small, fixed-size) cliques in graphs with labelled vertices representing probe-specific points of high interaction energy, and edges between a pair of vertices are labelled by the three-dimensional distance between the corresponding points of interaction. Our interest is in the discovery of frequent cliques that satisfy domain-specific constraints. In the paper, the discovery of such patterns is done using an Inductive Logic Programming (ILP) engine. The case for the use of ILP stems primarily from the explicit ways of incorporating domain-constraints, but any other technique capable of discovering frequent cliques from data can be used with some additional effort. The frequent cliques discovered are used to hypothesize pharmacophore-like structures on potential ligands. We test the utility of this approach by conducting a case study on the discovery of anti-malarials. Specifically, we test the approach on proteins belonging to the class of aspartic proteases. We are particularly interested in plasmepsin II, which is an enzyme in the haemoglobin degradation pathway of Plasmodium falciparum. We assess the pharmacophore-like constraints using: (a) a database of known inhibitors and non-inhibitors of aspartic proteases; and (b) a database of decoys that are physico-chemically similar to the aspartic proteases. Our results suggest that the approach could be used to obtain pharmacophores with good precision and recall for aspartic proteases.

Multi-Modal Clique-Graph Matching for View-Based 3D Model Retrieval

Multi-view matching is an important but a challenging task in view-based 3D model retrieval. To address this challenge, we propose an original multi-modal clique graph (MCG) matching method in this paper. We systematically present a method for MCG generation that is composed of cliques, which consist of neighbor nodes in multi-modal feature space and hyper-edges that link pairwise cliques. Moreover, we propose an image set-based clique/edgewise similarity measure to address the issue of the set-to-set distance measure, which is the core problem in MCG matching. The proposed MCG provides the following benefits: 1) preserves the local and global attributes of a graph with the designed structure; 2) eliminates redundant and noisy information by strengthening inliers while suppressing outliers; and 3) avoids the difficulty of defining high-order attributes and solving hyper-graph matching. We validate the MCG-based 3D model retrieval using three popular single-modal data sets and one novel multi-modal data set. Extensive experiments show the superiority of the proposed method through comparisons. Moreover, we contribute a novel real-world 3D object data set, the multi-view RGB-D object data set. To the best of our knowledge, it is the largest real-world 3D object data set containing multi-modal and multi-view information.

On self-clique shoal graphs

The clique graph of a graph G is the intersection graph K(G) of its (maximal) cliques, and G is self-clique if K(G) is isomorphic to G. A graph G is locallyH if the neighborhood of each vertex is isomorphic to H. Assuming that each clique of the regular and self-clique graph G is a triangle, it is known that G can only be r-regular for r∈{4,5,6} and G must be, depending on r, a locally H graph for some H∈{P4,P2∪P3,3P2}. The self-clique locally P4 graphs are easy to classify, but only a family of locally H self-clique graphs was known for H=P2∪P3, and another one for H=3P2.We study locally P2∪P3 graphs (i.e. shoal graphs). We show that all previously known shoal graphs were self-clique. We give a bijection from (finite) shoal graphs to 2-regular digraphs without directed 3-cycles. Under this translation, self-clique graphs correspond to self-dual digraphs, which simplifies constructions, calculations and proofs. We compute the numbers, for each n≤28, of self-clique and non-self-clique shoal graphs of order n, and also prove that these numbers grow at least exponentially with n.

Sparse cliques trump scale-free networks in coordination and competition.

Cooperative behavior, a natural, pervasive and yet puzzling phenomenon, can be significantly enhanced by networks. Many studies have shown how global network characteristics affect cooperation; however, it is difficult to understand how this occurs based on global factors alone, low-level network building blocks, or motifs are necessary. In this work, we systematically alter the structure of scale-free and clique networks and show, through a stochastic evolutionary game theory model, that cooperation on cliques increases linearly with community motif count. We further show that, for reactive stochastic strategies, network modularity improves cooperation in the anti-coordination Snowdrift game and the Prisoner’s Dilemma game but not in the Stag Hunt coordination game. We also confirm the negative effect of the scale-free graph on cooperation when effective payoffs are used. On the flip side, clique graphs are highly cooperative across social environments. Adding cycles to the acyclic scale-free graph increases cooperation when multiple games are considered; however, cycles have the opposite effect on how forgiving agents are when playing the Prisoner’s Dilemma game.

Rainbow cliques in edge-colored graphs

An edge-colored graph H is called rainbow if e(H)=c(H), where e(H) and c(H) are the number of edges of H and colors used in H, respectively. For two graphs G and H, the rainbow number rb(G,H) is the minimum number of colors k such that for every edge-coloring of G using k colors, G contains a rainbow H. In this paper we prove that for an edge-colored graph G on n vertices with n≥k≥4, if e(G)+c(G)≥(n2)+tn,k−2+2, then G contains a rainbow clique Kk, where tn,k−2 is the Turán number. This implies the known result rb(Kn,Kk)=tn,k−2+2, and moreover, rb(G,Kk)≤e(G¯)+rb(Kn,Kk) for n≥k≥4.