Complete Subgraphs Research Articles

Exploring multi-granularity contextual semantics for fully inductive knowledge graph completion

Fully inductive knowledge graph completion (KGC) aims to predict triplets involving both unseen entities and relations. Recent several approaches transform paths between entities into descriptions and modeling semantic correlations between paths using pre-trained language models (PLMs), have emerged as a promising solution for fully inductive reasoning. However, these methods often adopt a simplistic concatenation strategy for path-to-sentence transformation, which impedes PLMs’ ability to capture subtle nuances in context, resulting in sub-optimal path context embeddings. Furthermore, they ignore the high-order semantics underlying the complete context, which can provide richer information for inductive reasoning. To address these issues, we propose a Multi-Granularity Contextual Semantic (MGCS) modeling framework, utilizing a Path Modeling Network (PMN) and a Subgraph Modeling Network (SMN) to extract two granularity levels of contextual semantics from single paths and complete subgraphs, for fully inductive KGC. The PMN extracts paths between head and tail entities and employs reasoning patterns from similar cases to filter out unreliable paths. Then two innovative path conversion strategies are designed to significantly enhance the pre-trained language model’s understanding of specific path contexts. The SMN employs a neighbor interactive graph neural network to extract high-order semantics from the complete subgraph context with a concept-enhanced relation encoding, and optimizes it through a contrastive learning method. Finally, the confidence of the triples is evaluated from the perspective of global complete context by comparing the semantics between the subgraphs surrounding the target triplet and the subgraphs surrounding similar cases. Experimental results on benchmark datasets demonstrate the effectiveness of MGCS.

Characterising clique convergence for locally cyclic graphs of minimum degree δ ≥ 6

The clique graph kG of a graph G has as its vertices the cliques (maximal complete subgraphs) of G, two of which are adjacent in kG if they have non-empty intersection in G. We say that G is clique convergent if knG≅kmG for some n≠m, and that G is clique divergent otherwise. We completely characterise the clique convergent graphs in the class of (not necessarily finite) locally cyclic graphs of minimum degree δ≥6, showing that for such graphs clique divergence is a global phenomenon, dependent on the existence of large substructures. More precisely, we establish that such a graph is clique divergent if and only if its universal triangular cover contains arbitrarily large members from the family of so-called “triangular-shaped graphs”.

Orbits of finite-field hypergeometric functions and complete subgraphs of generalized Paley graphs

Orbits of finite-field hypergeometric functions and complete subgraphs of generalized Paley graphs

Katětov order between Hindman, Ramsey and summable ideals

A family I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {I}$$\\end{document} of subsets of a set X is an ideal on X if it is closed under taking subsets and finite unions of its elements. An ideal I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {I}$$\\end{document} on X is below an ideal J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {J}$$\\end{document} on Y in the Katětov order if there is a function f:Y→X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f{: }Y\\rightarrow X$$\\end{document} such that f-1[A]∈J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f^{-1}[A]\\in \\mathcal {J}$$\\end{document} for every A∈I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A\\in \\mathcal {I}$$\\end{document}. We show that the Hindman ideal, the Ramsey ideal and the summable ideal are pairwise incomparable in the Katětov order, whereThe Ramsey ideal consists of those sets of pairs of natural numbers which do not contain a set of all pairs of any infinite set (equivalently do not contain, in a sense, any infinite complete subgraph),The Hindman ideal consists of those sets of natural numbers which do not contain any infinite set together with all finite sums of its members (equivalently do not contain IP-sets that are considered in Ergodic Ramsey theory),The summable ideal consists of those sets of natural numbers such that the series of the reciprocals of its members is convergent.

KD-Club: An Efficient Exact Algorithm with New Coloring-Based Upper Bound for the Maximum k-Defective Clique Problem

The Maximum k-Defective Clique Problem (MDCP) aims to find a maximum k-defective clique in a given graph, where a k-defective clique is a relaxation clique missing at most k edges. MDCP is NP-hard and finds many real-world applications in analyzing dense but not necessarily complete subgraphs. Exact algorithms for MDCP mainly follow the Branch-and-bound (BnB) framework, whose performance heavily depends on the quality of the upper bound on the cardinality of a maximum k-defective clique. The state-of-the-art BnB MDCP algorithms calculate the upper bound quickly but conservatively as they ignore many possible missing edges. In this paper, we propose a novel CoLoring-based Upper Bound (CLUB) that uses graph coloring techniques to detect independent sets so as to detect missing edges ignored by the previous methods. We then develop a new BnB algorithm for MDCP, called KD-Club, using CLUB in both the preprocessing stage for graph reduction and the BnB searching process for branch pruning. Extensive experiments show that KD-Club significantly outperforms state-of-the-art BnB MDCP algorithms on the number of solved instances within the cut-off time, having much smaller search tree and shorter solving time on various benchmarks.

Nourishing Number of Some Associated Graphs

Let N0 = N∪{0} and P(N0) be the power set. An injection f : V (G) → P(N0) is an integer additive set-indexer (IASI) of a graph G if the induced map f+ : E(G) → P(N0) given by f+(uv) = f(u) + f(v) is also an injection, where f(u) + f(v) is the sumset of f(u) and f(v). Moreover, if |f+(uv)| = |f(u)| |f(v)|, for all uv in E(G), then f is a strong IASI of G. The nourishing number of a graph G is the minimum order of the maximal complete subgraph of G such that G admits a strong IASI. In this paper we investigate the admissibility of strong IASI for some associated graphs and calculate their nourishing number. In addition, we obtain the nourishing number of powers of the associated graphs.

A critical probability for biclique partition of Gn,p

The biclique partition number of a graph G=(V,E), denoted bp(G), is the minimum number of pairwise edge disjoint complete bipartite subgraphs of G so that each edge of G belongs to exactly one of them. It is easy to see that bp(G)≤n−α(G), where α(G) is the maximum size of an independent set of G. Erdős conjectured in the 80's that for almost every graph G equality holds; i.e., if G=Gn,1/2 then bp(G)=n−α(G) with high probability. Alon showed that this is false. We show that the conjecture of Erdős is true if we instead take G=Gn,p, where p is constant and less than a certain threshold value p0≈0.312. This verifies a conjecture of Chung and Peng for these values of p. We also show that if p0<p<1/2 then bp(Gn,p)=n−(1+Θ(1))α(Gn,p) with high probability.

An optimization algorithm for maximum quasi-clique problem based on information feedback model.

The maximum clique problem in graph theory is a well-known challenge that involves identifying the complete subgraph with the highest number of nodes in a given graph, which is a problem that is hard for nondeterministic polynomial time (NP-hard problem). While finding the exact application of the maximum clique problem in the real world is difficult, the relaxed clique model quasi-clique has emerged and is widely applied in fields such as bioinformatics and social network analysis. This study focuses on the maximum quasi-clique problem and introduces two algorithms, NF1 and NR1. These algorithms make use of previous iteration information through an information feedback model, calculate the information feedback score using fitness weighting, and update individuals in the current iteration based on the benchmark algorithm and selected previous individuals. The experimental results from a significant number of composite and real-world graphs indicate that both algorithms outperform the original benchmark algorithm in dense instances, while also achieving comparable results in sparse instances.

From Stars to Diamonds: Counting and Listing Almost Complete Subgraphs in Large Networks

Abstract Listing dense subgraphs is a fundamental task with a variety of network analytics applications. A lot of research has been done focusing on $k$-cliques, i.e. complete subgraphs on $k$ nodes. However, requiring complete connectivity between the nodes of a subgraph may be too restrictive in many real applications. Hence, in this paper, we consider a natural relaxation of cliques, called $k$-diamonds and defined as cliques of size $k$ with one missing edge. We first provide a sequential algorithm that, in $O(nm^{(k-1)/2})$ time, counts and lists all the $k$-diamonds in large graphs, for any constant $k \geq 4$. A parallel extension of the sequential algorithm is then proposed and analyzed in a MapReduce-style model, achieving the same local and total space usage of the state-of-the-art algorithms for $k$-cliques. The running time is optimal on dense graphs and $O(\sqrt{m})$ larger than $k$-clique counting if the graph is sparse. Our algorithms compute induced diamonds by analyzing the structure of directed stars formed by the graph nodes and their neighbors.

Perfect state transfer by means of discrete-time quantum walk on the complete bipartite graph

Perfect state transfer has attracted a great deal of attention recently due to its crucial role in quantum communication and scalable quantum computation. In this paper, we propose the perfect state transfer algorithms with a pair of sender-receiver and two pairs of sender-receiver on the complete bipartite graph respectively. The algorithm with a pair of sender-receiver is implemented through discrete-time quantum walk, flexibly setting the coin operators based on the positions of the sender and receiver. The algorithm with two pairs of sender-receiver ensures that the two quantum states are distributed on both sides of the complete bipartite graph during the process, thereby achieving perfect state transfer. In addition, the quantum circuits corresponding to the algorithms are provided. The algorithms can transfer an arbitrary quantum state and can simultaneously transfer two arbitrary quantum states from the senders to the receivers in any case. Moreover, the algorithms are not only applicable to complete bipartite graphs but also to more graph structures with complete bipartite subgraphs, which will provide potential applications for quantum information processing.

Exact values for some unbalanced Zarankiewicz numbers

AbstractFor positive integers , , and , the Zarankiewicz number is defined to be the maximum number of edges in a bipartite graph with parts of sizes and that has no complete bipartite subgraph containing vertices in the part of size and vertices in the part of size . A simple argument shows that, for each , when . Here, for large , we determine the exact value of in almost all of the remaining cases where . We establish a new family of upper bounds on which complement a family already obtained by Roman. We then prove that the floor of the best of these bounds is almost always achieved. We also show that there are cases in which this floor cannot be achieved and others in which determining whether it is achieved is likely a very hard problem. Our results are proved by viewing the problem through the lens of linear hypergraphs and our constructions make use of existing results on edge decompositions of dense graphs.

Properly colored cycles in edge-colored 2-colored-triangle-free complete graphs

An edge-colored graph Gc is called properly colored if any two adjacent edges receive distinct colors. An edge-colored graph Gc is 2-colored-triangle-free if Gc contains no 2-colored-triangle, where a 2-colored-triangle is an edge-colored triangle with exactly two colors. Let dc(v) be the number of colors on the edges incident to v in Gc and let δc(Gc) be the minimum dc(v) for all v∈V(Gc). In this paper we extend the definitions of proper vertex-pancyclic and proper edge-pancyclic to proper k-path-pancyclic, defined as follows: An edge-colored graph Gc is said to be proper k-path-pancyclic if each properly colored path of length k is contained in a properly colored cycle of length l for every l with max{3,k+2}≤l≤|V(Gc)|. We prove that an edge-colored 2-colored-triangle-free complete graph Gc with δc(Gc)≥k+3 is either (almost) proper k-path-pancyclic or contains a large monochromatic complete subgraph.

Wireless random‐access networks with bipartite interference graphs

AbstractWe consider random‐access networks where nodes represent servers with a queue and can be either active or inactive. A node deactivates at unit rate, while it activates at a rate that depends on its queue length, provided none of its neighbors is active. We consider arbitrary bipartite graphs in the limit as the initial queue lengths become large and identify the transition time between the two states where one half of the network is active and the other half is inactive. The transition path is decomposed into a succession of transitions on complete bipartite subgraphs. We formulate a randomized greedy algorithm that takes the graph as input and gives as output the set of transition paths the network is most likely to follow. Along each path we determine the mean transition time and its law on the scale of its mean. Depending on the activation rates, we identify three regimes of behavior.

Polynomial bounds for chromatic number. V. Excluding a tree of radius two and a complete multipartite graph

The Gyárfás-Sumner conjecture says that for every forest H and every integer k, if G is H-free and does not contain a clique on k vertices then it has bounded chromatic number. (A graph is H-free if it does not contain an induced copy of H.) Kierstead and Penrice proved it for trees of radius at most two, but otherwise the conjecture is known only for a few simple types of forest. More is known if we exclude a complete bipartite subgraph instead of a clique: Rödl showed that, for every forest H, if G is H-free and does not contain Kt,t as a subgraph then it has bounded chromatic number. In an earlier paper with Sophie Spirkl, we strengthened Rödl's result, showing that for every forest H, the bound on chromatic number can be taken to be polynomial in t. In this paper, we prove a related strengthening of the Kierstead-Penrice theorem, showing that for every tree H of radius two and integer d≥2, if G is H-free and does not contain as a subgraph the complete d-partite graph with parts of cardinality t, then its chromatic number is at most polynomial in t.

On some conjectures on biclique graphs

The bicliqueB of a graph G is a maximal induced complete bipartite subgraph of G. The biclique graphKB(G) of a graph G is the graph with the bicliques of G as its vertices where two vertices of KB(G) are adjacent if and only if they have a common vertex in G. In this paper, we disprove a conjecture of Groshaus and Montero concerning Helly property of biclique graphs (Groshaus and Montero, 2021) and two conjectures of Montero concerning vertex removal in biclique graphs (Montero, 2022). In addition, we confirm a conjecture of Montero about the structural characterization of biclique graphs in the same paper.

Elusive properties of infinite graphs

AbstractA graph property is said to be elusive (or evasive) if every algorithm testing this property by asking questions of the form “is there an edge between vertices and ?” requires, in the worst case, to ask about all pairs of vertices. The unsettled Aanderaa–Karp–Rosenberg conjecture is that every nontrivial monotone graph property is elusive for any finite vertex set. We show that the situation is completely different for infinite vertex sets: the monotone graph properties “every vertex has degree at least ” and “every connected component has size at least ,” where and are natural numbers, are not elusive for infinite vertex sets, but the monotone graph property “the graph contains a cycle” is elusive for arbitrary vertex set. On the other hand, we also prove that every algorithm testing some natural monotone graph properties, for example, “every vertex has degree at least ” or “connected” on the vertex set should check “lots of edges,” more precisely, all the edges of an infinite complete subgraph.

Structural properties of graph products

AbstractDujmovć, Joret, Micek, Morin, Ueckerdt, and Wood established that every planar graph is a subgraph of the strong product of a graph with bounded treewidth and a path. Motivated by this result, this paper systematically studies various structural properties of cartesian, direct and strong products. In particular, we characterise when these graph products contain a given complete multipartite subgraph, determine tight bounds for their degeneracy, establish new lower bounds for the treewidth of cartesian and strong products, and characterise when they have bounded treewidth and when they have bounded pathwidth.

On Orthogonal Double Covers and Decompositions of Complete Bipartite Graphs by Caterpillar Graphs

Nowadays, graph theory is one of the most exciting fields of mathematics due to the tremendous developments in modern technology, where it is used in many important applications. The orthogonal double cover (ODC) is a branch of graph theory and is considered as a special class of graph decomposition. In this paper, we decompose the complete bipartite graphs Kx,x by caterpillar graphs using the method of ODCs. The article also deals with constructing the ODCs of Kx,x by general symmetric starter vectors of caterpillar graphs such as stars–caterpillar, the disjoint copies of cycles–caterpillars, complete bipartite caterpillar graphs, and the disjoint copies of caterpillar paths. We decompose the complete bipartite graph by the complete bipartite subgraphs and by the disjoint copies of complete bipartite subgraphs using general symmetric starter vectors. The advantage of some of these new results is that they enable us to decompose the giant networks into large groups of small networks with the comprehensive coverage of all parts of the giant network by using the disjoint copies of symmetric starter subgraphs. The use case of applying the described theory for various applications is considered.

Unavoidable patterns in locally balanced colourings

AbstractWhich patterns must a two-colouring of $K_n$ contain if each vertex has at least $\varepsilon n$ red and $\varepsilon n$ blue neighbours? We show that when $\varepsilon \gt 1/4$ , $K_n$ must contain a complete subgraph on $\Omega (\log n)$ vertices where one of the colours forms a balanced complete bipartite graph.When $\varepsilon \leq 1/4$ , this statement is no longer true, as evidenced by the following colouring $\chi$ of $K_n$ . Divide the vertex set into $4$ parts nearly equal in size as $V_1,V_2,V_3, V_4$ , and let the blue colour class consist of the edges between $(V_1,V_2)$ , $(V_2,V_3)$ , $(V_3,V_4)$ , and the edges contained inside $V_2$ and inside $V_3$ . Surprisingly, we find that this obstruction is unique in the following sense. Any two-colouring of $K_n$ in which each vertex has at least $\varepsilon n$ red and $\varepsilon n$ blue neighbours (with $\varepsilon \gt 0$ ) contains a vertex set $S$ of order $\Omega _{\varepsilon }(\log n)$ on which one colour class forms a balanced complete bipartite graph, or which has the same colouring as $\chi$ .

Computing Significant Cliques in Large Labeled Networks

Mining cohesive subgraphs and communities is a fundamental problem in network analysis and has drawn much attention in the last decade. Most existing cohesive subgraph models mainly consider the structural cohesion but ignore the subgraph significance. In this paper, we formulate a new model, called statistically significant clique, to mine significant cohesive subgraphs in large vertex-labeled graphs. A statistically significant clique is a complete subgraph with a significance value exceeding a given threshold. The subgraph significance is evaluated by a widely used metric called chi-square statistic. We study the problem of enumerating all maximal statistically significant cliques. The problem is proved to be NP-hard. We propose an efficient branch-and-bound algorithm with several elegant pruning strategies to solve our problem. We conduct extensive experiments on seven large real-world datasets to show the practical efficiency of our algorithms. We also conduct a case study to evaluate the effectiveness of our proposed model.