Clique Size Research Articles

The complexity of subtree intersection representation of chordal graphs and linear time chordal graph generation

It is known that any chordal graph on n vertices can be represented as the intersection of n subtrees in a tree on n nodes (Gavril in J Comb Theory 16:47–56, 1974). This characterization has been recently used to generate random chordal graphs on n vertices by generating n subtrees of a tree on n nodes. The space (and thus time) complexity of an algorithm generating n subtrees of a tree on n nodes is at least the sum of the sizes of the generated subtrees. The determination of this sum was left as an open question in Şeker et al. (Generation of random chordal graphs using subtrees of a tree. arXiv preprint arXiv:1810.13326, 2018). In this paper, we show that the sum of the sizes of n subtrees in a tree on n nodes is \(\varTheta (m\sqrt{n})\). We also show that we can confine ourselves to contraction-minimal subtree intersection representations because they are sufficient to generate every chordal graph with strictly positive probability. Moreover, the sum of the sizes of the subtrees in a contraction-minimal representation is at most \(2m+n\). We use this result to derive the first linear-time random chordal graph generator. Based on contraction-minimal representations, we also derive connectivity-related structural properties of chordal graphs. Besides these theoretical results, we also conduct experiments to study the quality of the chordal graphs generated by our algorithm and compare them to those generated by existing methods from the literature. Our algorithm does not generate chordal graphs uniformly at random, which is a quite challenging open question, irrespective of the time complexity of the generator. However, our experimental study suggests that the generated graphs have a fairly varied structure as indicated by the sizes of maximal cliques. Furthermore, our algorithm is simple to implement and produces graphs with 10,000 vertices and \(4 \times 10^7\) edges in less than one second on a laptop computer.

Improved hardness and approximation results for single allocation hub location problems

Given a metric graph G=(V,E,w) and an integer k, we aim to find a single allocation k-hub location, which is a spanning subgraph consisting of a clique of size k such that every node outside of the clique is adjacent to exactly one node inside the clique. For various objective functions studied in the literature, we present improved hardness and approximation results.

Many Cliques with Few Edges

Recently Cutler and Radcliffe proved that the graph on $n$ vertices with maximum degree at most $r$ having the most cliques is a disjoint union of $\lfloor n/(r+1)\rfloor$ cliques of size $r+1$ together with a clique on the remainder of the vertices. It is very natural also to consider this question when the limiting resource is edges rather than vertices. In this paper we prove that among graphs with $m$ edges and maximum degree at most $r$, the graph that has the most cliques of size at least two is the disjoint union of $\bigl\lfloor m \bigm/\binom{r+1}{2} \bigr\rfloor$ cliques of size $r+1$ together with the colex graph using the remainder of the edges.

Graph classes with linear Ramsey numbers

The Ramsey number RX(p,q) for a class of graphs X is the minimum n such that every graph in X with at least n vertices has either a clique of size p or an independent set of size q. We say that Ramsey numbers are linear in X if there is a constant k such that RX(p,q)≤k(p+q) for all p,q. In the present paper we conjecture that if X is a hereditary class defined by finitely many forbidden induced subgraphs, then Ramsey numbers are linear in X if and only if X excludes a forest, a disjoint union of cliques and their complements. We prove the “only if” part of this conjecture and verify the “if” part for a variety of classes. We also apply the notion of linearity to bipartite Ramsey numbers and reveal a number of similarities and differences between the bipartite and non-bipartite case.

A branch-and-cut algorithm for the Edge Interdiction Clique Problem

Given a graph G and an interdiction budget k∈N, the Edge Interdiction Clique Problem (EICP) asks to find a subset of at most k edges to remove from G so that the size of the maximum clique, in the interdicted graph, is minimized. The EICP belongs to the family of interdiction problems with the aim of reducing the clique number of the graph. The EICP optimal solutions, called optimal interdiction policies, determine the subset of most vital edges of a graph which are crucial for preserving its clique number. We propose a new set-covering-based Integer Linear Programming (ILP) formulation for the EICP with an exponential number of constraints, called the clique-covering inequalities. We design a new branch-and-cut algorithm which is enhanced by a tailored separation procedure and by an effective heuristic initialization phase. Thanks to the new exact algorithm, we manage to solve the EICP in several sets of instances from the literature. Extensive tests show that the new exact algorithm greatly outperforms the state-of-the-art approaches for the EICP.

On the complexity of coloring ‐graphs

AbstractAn ‐graph is a graph that can be partitioned into r independent sets and ℓ cliques. In the Graph Coloring problem, we are given as input a graph G, and the objective is to determine the minimum integer k such that G admits a proper vertex k‐coloring. In this work, we describe a Poly versus NP‐hard dichotomy of this problem regarding the parameters r and ℓ of ‐graphs, which determine the boundaries of the ‐hardness of Graph Coloring for such classes. We also analyze the complexity of the problem on ()‐graphs under the parameterized complexity perspective. We show that given a (2, 1)‐partition of a graph G, finding an optimal coloring of G is ‐complete even when K1 is a maximal clique of size 3; XP but W[1]‐hard when parameterized by ; fixed‐parameter tractable (FPT) and admits a polynomial kernel when parameterized by . Besides, concerning the case where K1 is a maximal clique of size 3, a P versus NPh dichotomy regarding the neighborhood of K1 is provided; furthermore, an FPT algorithm parameterized by the number of vertices having no neighbor in K1 is presented.

Kernel for Kt-free Edge Deletion

In the Kt-free Edge Deletion problem the input is a graph G and an integer k, and the goal is to decide whether there is a set of at most k edges of G whose removal results in a graph with no clique of size t. In this paper we give a kernel to this problem with O(kt−1) vertices and edges.

Kr-factors in graphs with low independence number

A classical result by Hajnal and Szemerédi from 1970 determines the minimal degree conditions necessary to guarantee for a graph to contain a Kr-factor. Namely, any graph on n vertices, with minimum degree δ(G)≥(1−1r)n and r dividing n has a Kr-factor. This result is tight but the extremal examples are unique in that they all have a large independent set which is the bottleneck. Nenadov and Pehova showed that by requiring a sub-linear independence number the minimum degree condition in the Hajnal-Szemerédi theorem can be improved. We show that, with the same minimum degree and sub-linear independence number, we can find a clique-factor with double the clique size. More formally, we show for every r∈N and constant μ>0 there is a positive constant γ such that every graph G on n vertices with δ(G)≥(1−2r+μ)n and α(G)<γn has a Kr-factor. We also give examples showing the minimum degree condition is asymptotically best possible.

Percolation on interdependent networks with cliques and weak interdependence

Previous researches on cascading failures on interdependent networks with cliques mainly consider “strong” interdependence, that is, once a clique fails, its dependency partner in the other layer will fail completely. However, many interdependent systems in real-world may have self-sustaining abilities, and the failure of one clique may only destroy part of the function of its dependency partner, which can be called “weak” interdependence. In this paper, based on the framework of percolation, a cascading failure model on the interdependent networks with cliques and weak interdependence is investigated analytically and numerically. The results of extensive simulations show that the type of the phase transitions of both layers can be altered from discontinuous phase transitions to continuous phase transitions with the decrease of interdependence strength. In addition, in the process from discontinuous phase transitions to continuous phase transitions, the whole system exhibits a surprising phenomenon of mixed phase transitions: layer with large size of cliques percolates continuously while layer with small size of cliques percolates discontinuously. It indicates that weak interdependence strength and the structure of cliques can enhance the robustness of the interdependent networks. The theoretical and numerical predictions agree well with each other.

Cliques in high-dimensional random geometric graphs

Random geometric graphs have become now a popular object of research. Defined rather simply, these graphs describe real networks much better than classical Erdős–Rényi graphs due to their ability to produce tightly connected communities. The n vertices of a random geometric graph are points in d-dimensional Euclidean space, and two vertices are adjacent if they are close to each other. Many properties of these graphs have been revealed in the case when d is fixed. However, the case of growing dimension d is practically unexplored. This regime corresponds to a real-life situation when one has a data set of n observations with a significant number of features, a quite common case in data science today. In this paper, we study the clique structure of random geometric graphs when nrightarrow infty, and d rightarrow infty, and average vertex degree grows significantly slower than n. We show that under these conditions, random geometric graphs do not contain cliques of size 4 a. s. if only d gg log ^{1 + epsilon } n. As for the cliques of size 3, we present new bounds on the expected number of triangles in the case log ^2 n ll d ll log ^3 n that improve previously known results. In addition, we provide new numerical results showing that the underlying geometry can be detected using the number of triangles even for small n.

Indicators for assessing the robustness of metapopulations against habitat loss

Habitat loss and fragmentation resulting from environmental changes are main drivers of global biodiversity loss, as the survival of metapopulations relies on the ability of individuals to disperse among suitable habitat patches. To prioritize conservation efforts, methods are needed for evaluating the robustness of metapopulations against habitat loss. We therefore investigate this robustness for different degrees of habitat loss, for different types of habitat loss (random, peripheral, and contagious) and of habitat networks, and for species differing in their local-extinction risks and dispersal ranges. In particular, we analyse several standard network types (with random, regular, small-world, or scale-free structure) and compare them with several alternative network types derived from real-world two-dimensional habitat landscapes (with random, clustered, or contiguous habitat allocation). Furthermore, we investigate how well 29 different graph-theoretic metrics of habitat networks can serve as indicators of metapopulation robustness against habitat loss – as this approach, where feasible, allows replacing complex simulation-based predictions with simple indicator-based predictions. We find that responses of species to habitat loss on the considered landscape-based habitat networks qualitatively differ from those on the considered standard habitat networks. This suggests that results obtained for the latter, albeit widely examined in the literature, can be unrepresentative and misleading. As expected, species with high risks of local extinction and short dispersal ranges are particularly vulnerable to habitat loss, across all considered types of habitat loss and habitat networks. The graph-theoretic network metric that best explains the robustness of metapopulations against habitat loss depends on the considered types of species, habitat networks, and habitat loss. None of the examined metrics give consistently reliable predictions under all circumstances. For sensitive species, characterized by high local-extinction risks and short dispersal ranges, a network’s average clique size, redundancy, average degree, connectance, clustering coefficient, and average closeness centrality are the best indicators of metapopulation robustness. For landscape-based habitat networks, a network’s average clique size, beta coefficient, clustering coefficient, redundancy, and cyclomatic number work best. For contagious habitat loss, the network type has a particularly strong impact on the species-specific robustness against habitat loss. In summary, our study introduces a method for evaluating the robustness of metapopulations against habitat loss and shows that a network’s clustering coefficient, under a wide range of circumstances, is a particularly reliable indicator of this robustness.

Equitable Clique-Coloring in Claw-Free Graphs with Maximum Degree at Most 4

A clique of a graph G is a set of pairwise adjacent vertices of G. A clique-coloring of G is an assignment of colors to the vertices of G in such a way that no inclusion-wise maximal clique of size at least two of G is monochromatic. An equitable clique-coloring of G is a clique-coloring such that any two color classes differ in size by at most one. Bacso and Tuza proved that connected claw-free graphs with maximum degree at most four, other than chordless odd cycles of order greater than three, are 2-clique-colorable and a 2-clique-coloring can be found in $$O(n^{2})$$ Bacso and Tuza (Discrete Math Theor Comput Sci 11(2):15–24, 2009). In this paper we prove that every connected claw-free graph with maximum degree at most four, not a chordless odd cycle of order greater than three, has an equitable 2-clique-coloring. In addition we improve the algorithm described in the paper mentioned by giving an equitable 2-clique-coloring in linear time for this class of graphs.

Additive non-approximability of chromatic number in proper minor-closed classes

Robin Thomas asked whether for every proper minor-closed class G, there exists a polynomial-time algorithm approximating the chromatic number of graphs from G up to a constant additive error independent on the class G. We show this is not the case: unless P=NP, for every integer k≥1, there is no polynomial-time algorithm to color a K4k+1-minor-free graph G using at most χ(G)+k−1 colors. More generally, for every k≥1 and 1≤β≤4/3, there is no polynomial-time algorithm to color a K4k+1-minor-free graph G using less than βχ(G)+(4−3β)k colors. As far as we know, this is the first non-trivial non-approximability result regarding the chromatic number in proper minor-closed classes.Furthermore, we give somewhat weaker non-approximability bound for K4k+1-minor-free graphs with no cliques of size 4. On the positive side, we present additive approximation algorithm whose error depends on the apex number of the forbidden minor, and an algorithm with additive error 6 under the additional assumption that the graph has no 4-cycles.

The maximum number of triangles in a graph of given maximum degree

Maximizing or minimizing the number of copies of a fixed graph in a large host graph is one of the most classical topics in extremal graph theory. Indeed, one of the most famous problems in extremal graph theory, the Erdős-Rademacher problem, which can be traced back to the 1940s, asks to determine the minimum number of triangles in a graph with a given number of vertices and edges. It was conjectured that the mnimum is attained by complete multipartite graphs with all parts but one of the same size whilst the remaining part may be smaller. The problem was widely open in the regime of four or more parts until Razborov resolved the problem asymptotically in 2008 as one of the first applications of his newly developed flag algebra method. This catalyzed a line of research on the structure of extremal graphs and extensions. In particular, Reiher asymptotically solved in 2016 the conjecture of Lovász and Simonovits from the 1970s that the same graphs are also minimizers for cliques of arbitrary size. This paper deals with a problem concerning the opposite direction: _What is the maximum number of triangles in a graph with a given number $n$ of vertices and a given maximum degree $D$?_ Gan, Loh and Sudakov in 2015 conjectured that the graph maximizing the number of triangles is always a union of disjoint cliques of size $D+1$ and another clique that may be smaller, and showed that if such a graph maximizes the number of triangles, it also maximizes the number of cliques of any size $r\ge 4$. The author presents a remarkably simple and elegant argument that proves the conjecture exactly for all $n$ and $D$.

Maximizing the number of cliques in graphs with given matching number

A matching M in a graph G is a set of disjoint edges of G. The matching number ν(G) is the number of edges in a maximum matching of G. The number of cliques of size s in G is denoted by Ns(G). Given positive integers n, k, and r, the graph H(n,k,r) is the one obtained from K2k+1−r by connecting each vertex of an independent set of size n−(2k+1−r) to the same r vertices chosen from K2k+1−r. Set hs(n,k,r)=Ns(H(n,k,r))=2k+1−rs+(n−2k−1+r)rs−1. In 1959, Erdős and Gallai proved that if G is a graph with n≥2k+2 vertices and ν(G)≤k, then e(G)≤max{2k+12,h2(n,k,k)} and characterized the extremal graphs. In this paper, motivated by a question proposed by Füredi, Kostochka, and Luo, we first extend Erdős and Gallai’s result. To be precisely, we prove that if G is a graph with n≥2k+2 vertices, minimum degree δ, and ν(G)≤k, then Ns(G)≤max{hs(n,k,δ),hs(n,k,k)} for each s≥2. Moreover, we are able to prove stability versions of this result specialized to edges. Namely, we prove if e(G)>max{h2(n,k,δ),h2(n,k,k−2)} and n≥2k+2, then G is a subgraph of H(n,k,k) or H(n,k,k−1). If e(G)>max{h2(n,k,δ+2),h2(n,k,k)} and n≥2k+11, then G is a subgraph of H(n,k,δ) or H(n,k,δ+1).

Improved Ramsey-type results for comparability graphs

AbstractSeveral discrete geometry problems are equivalent to estimating the size of the largest homogeneous sets in graphs that happen to be the union of few comparability graphs. An important observation for such results is that if G is an n-vertex graph that is the union of r comparability (or more generally, perfect) graphs, then either G or its complement contains a clique of size $n^{1/(r+1)}$ .This bound is known to be tight for $r=1$ . The question whether it is optimal for $r\ge 2$ was studied by Dumitrescu and Tóth. We prove that it is essentially best possible for $r=2$ , as well: we introduce a probabilistic construction of two comparability graphs on n vertices, whose union contains no clique or independent set of size $n^{1/3+o(1)}$ .Using similar ideas, we can also construct a graph G that is the union of r comparability graphs, and neither G nor its complement contain a complete bipartite graph with parts of size $cn/{(log n)^r}$ . With this, we improve a result of Fox and Pach.

Temporal locality-aware sampling for accurate triangle counting in real graph streams

If we cannot store all edges in a dynamic graph, which edges should we store to estimate the triangle count accurately? Counting triangles (i.e., cliques of size three) is a fundamental graph problem with many applications in social network analysis, web mining, anomaly detection, etc. Recently, much effort has been made to accurately estimate the counts of global triangles (i.e., all triangles) and local triangles (i.e., all triangle incident to each node) in large dynamic graphs, especially with limited space. Although existing algorithms use sampling techniques without considering temporal dependencies in edges, we observe temporal locality in the formation of triangles in real dynamic graphs. That is, future edges are more likely to form triangles with recent edges than with older edges. In this work, we propose a family of single-pass streaming algorithms called Waiting-Room Sampling (WRS) for estimating the counts of global and local triangles in a fully dynamic graph, where edges are inserted and deleted over time, within a fixed memory budget. WRS exploits the temporal locality by always storing the most recent edges, which future edges are more likely to form triangles with, in the waiting room, while it uses reservoir sampling and its variant for the remaining edges. Our theoretical and empirical analyses show that WRS is: (a) Fast and ‘any time’: runs in linear time, always maintaining and updating estimates, while the input graph evolves, (b) Effective: yields up to 47% smaller estimation error than its best competitors, and (c) Theoretically sound: gives unbiased estimates with small variances under the temporal locality.

Development of Stock Networks Using Part Mutual Information and Australian Stock Market Data.

Complex network is a powerful tool to discover important information from various types of big data. Although substantial studies have been conducted for the development of stock relation networks, correlation coefficient is dominantly used to measure the relationship between stock pairs. Information theory is much less discussed for this important topic, though mutual information is able to measure nonlinear pairwise relationship. In this work we propose to use part mutual information for developing stock networks. The path-consistency algorithm is used to filter out redundant relationships. Using the Australian stock market data, we develop four stock relation networks using different orders of part mutual information. Compared with the widely used planar maximally filtered graph (PMFG), we can generate networks with cliques of large size. In addition, the large cliques show consistency with the structure of industrial sectors. We also analyze the connectivity and degree distributions of the generated networks. Analysis results suggest that the proposed method is an effective approach to develop stock relation networks using information theory.

Minimizing the numbers of cliques and cycles of fixed size in an [formula omitted]-saturated graph

This paper considers two important questions in the well-studied theory of graphs that are F-saturated. A graph G is called F-saturated if G does not contain a subgraph isomorphic to F, but the addition of any edge creates a copy of F. We first resolve a fundamental question of minimizing the number of cliques of size r in a Ks-saturated graph for all sufficiently large numbers of vertices, confirming a conjecture of Kritschgau, Methuku, Tait, and Timmons. We also go further and prove a corresponding stability result. Next we minimize the number of cycles of length r in a Ks-saturated graph for all sufficiently large numbers of vertices, and classify the extremal graphs for most values of r, answering another question of Kritschgau, Methuku, Tait, and Timmons for most r.We then move on to a central and longstanding conjecture in graph saturation made by Tuza, which states that for every graph F, the limit limn→∞sat(n,F)n exists, where sat(n,F) denotes the minimum number of edges in an n-vertex F-saturated graph. Pikhurko made progress in the negative direction by considering families of graphs instead of a single graph, and proved that there exists a graph family F of size 4 for which limn→∞sat(n,F)n does not exist (for a family of graphs F, a graph G is called F-saturated if G does not contain a copy of any graph in F, but the addition of any edge creates a copy of a graph in F, and sat(n,F) is defined similarly). We make the first improvement in 15 years by showing that there exist infinitely many graph families of size 3 where this limit does not exist. Our construction also extends to the generalized saturation problem when we minimize the number of fixed-size cliques. We also show an example of a graph Fr for which there is irregular behavior in the minimum number of Cr’s in an n-vertex Fr-saturated graph.

Cliques and Chromatic Number in Multiregime Random Graphs

In this paper, we study cliques and chromatic number in the random subgraph Gn of the complete graph Kn on n vertices, where each edge is independently open with a probability pn. Associating Gn with the probability measure ℙn, we say that the sequence {ℙn} is multiregime if the edge probability sequence {pn} is not convergent. Using a recursive method we obtain uniform bounds on the maximum clique size and chromatic number for such multiregime random graphs.