Maximum Cardinality Research Articles

Semidefinite Programming Bounds for Spherical Three-Distance Sets

A spherical three-distance set is a finite collection $X$ of unit vectors in $\mathbb{R}^{n}$ such that the set of the distances between any two distinct vectors has cardinality three. We use the semidefinite programming method to improve the upper bounds for the cardinalities of spherical three-distance sets in the Euclidean spaces of several dimensions. We obtain better bounds in $\mathbb{R}^7$, $\mathbb{R}^{20}$, $\mathbb{R}^{21}$, $\mathbb{R}^{23}$, $\mathbb{R}^{24}$ and $\mathbb{R}^{25}$. In particular, we prove that the maximum cardinality of a spherical three-distance set in $\mathbb R^{23}$ is $2300$.

Spectrality and non-spectrality of a class of Moran measures

Let {Mn}n=1∞∈M2(R) be a sequence of real expanding matrices and {Dn}n=1∞ be a sequence of digit sets withDn={(00),(anbn),(cndn)}, where |andn−bncn|=ϕ(n)∈Z, and letμ{Mn},{Dn}=δM1−1D1⁎δ(M1M2)−1D2⁎δ(M1M2M3)−1D3⁎⋯ be the associated Moran measure. This paper establishes a necessary and sufficient condition for μMn,Dn to be a spectral measure when each Mn is a diagonal matrix, and characterizes the structure of the spectrum Λ. Furthermore, we demonstrate that for any sequence of matrices {Mn}n=1∞ ∈ M2(Z) with det⁡(Mn)∉3Z, the space L2(μ{Mn},{Dn}) contains at most 9 mutually orthogonal exponential functions. The precise maximum cardinality of such functions is also determined.

The upper geodetic vertex covering number of a graph

A set S ⊆ V (G) is a geodetic vertex cover of G if S is both a geodetic set and a vertex cover of G. The minimum cardinality of a geodetic vertex cover of G is defined as the geodetic vertex covering number of G and is denoted by gα(G) . A geodetic vertex cover S in a connected graph G is called a minimal geodetic vertex cover of G if no proper subset of S is a geodetic vertex cover of G. The upper geodetic vertex covering number g+­­α(G) of G is the maximum cardinality of a minimal geodetic vertex cover of G. Some general properties satisfied by the upper geodetic vertex covering number of a graph are studied. The upper geodetic vertex covering number of several classes of graphs are determined. Some bounds for g+­­α(G) are obtained and the graphs attaining these bounds are characterized.

Edge open packing: Complexity, algorithmic aspects, and bounds

Given a graph G, two edges e1,e2∈E(G) are said to have a common edge e≠e1,e2 if e joins an endvertex of e1 to an endvertex of e2. A subset B⊆E(G) is an edge open packing set in G if no two edges of B have a common edge in G, and the maximum cardinality of such a set in G is called the edge open packing number, ρeo(G), of G. In this paper, we prove that the decision version of the edge open packing number is NP-complete even when restricted to graphs with universal vertices, Eulerian bipartite graphs, and planar graphs with maximum degree 4, respectively. In contrast, we present a linear-time algorithm that computes the edge open packing number of a tree. We also resolve two problems posed in the seminal paper (Chelladurai et al. (2022) [5]). Notably, we characterize the graphs G that attain the upper bound ρeo(G)≤|E(G)|/δ(G), and provide lower and upper bounds for the edge-deleted subgraph of a graph and establish the corresponding realization result.

Maximum Cardinality \(f\) -Matching in Time \(O(n^{2/3}m)\)

We present an algorithm that finds a maximum cardinality \(f\) -matching of a simple graph in time \(O(n^{2/3}m)\) . Here \(f:V\to\mathbb{N}\) is a given function and an \(f\) -matching is a subgraph wherein each vertex \(v\in V\) has degree \(\leq f(v)\) . This result generalizes a string of algorithms that concentrate on simple bipartite graphs. The bipartite case is based on the notion of level graph, introduced by Dinic for network flow. In general graphs this notion breaks down: Vertices no longer have unique levels, and there are too many levels to analyze the corresponding level graph like bipartite graphs ( \(\Theta(n^{2})\) vs. \(n\) ). We use “natural” levels to prove properties of shortest augmenting trails (e.g., formulas for trail length). We use “shortened” levels to derive the algorithm's time bound. The algorithm, unmodified, is also efficient on multigraphs, achieving time \(O(\min\{\sqrt{f(V)},n\} m)\) for \(f(V)=\sum_{v}f(v)\) . The special case \(f\equiv 1\) shows the algorithm duplicates the classic time bound for maximum cardinality matching, \(O(\sqrt{n} m)\) .

Reliability Analysis of (n, k)-Bubble-Sort Networks Based on Extra Conditional Fault

Given a graph G=(V(G),V(E)), a non-negative integer g and a set of faulty vertices F⊆V(G), the g-extra connectivity of G, denoted by κg(G), is the smallest cardinality of F, whose value of deletion, if exists, will disconnect G and give each remaining component at least g+1 vertices. The g-extra diagnosability of the graph G, denoted by tg(G), is the maximum cardinality of the set F of fault vertices that the graph can guarantee to identify under the condition that each fault-free component has more than g vertices. In this paper, we determine that g-extra connectivity of (n,k)-bubble-sort network Bn,k is κg(Bn,k)=n+g(k−2)−1 for 4≤k≤n−1 and 0≤g≤n−k. Afterwards, we show that g-extra diagnosability of Bn,k under the PMC model (4≤k≤n−1 and 0≤g≤n−k) and MM* model (4≤k≤n−1 and 0≤g≤min{n−k−1,k−2}) is tg(Bn,k)=n+g(k−1)−1, respectively.

Some Results of Total Domatic Number on Anti Fuzzy Graph

Let AG = (N, A, σ, μ) be an anti fuzzy graph. A partition DP = {D1, D2, ….., DK} of N(AG) is referred to as total domatic partition of AG if for each Di is a total dominating set of anti fuzzy graph AG and N(AG)= ⋃Di. The maximum cardinality taken over all maximum number of classes with a minimal total domatic partition of AG is called the total domatic number of AG and it is denoted by . The maximum number of classes with maximum fuzzy cardinality of a partition Di (AG) is called anti fuzzy total domatic number of anti fuzzy graph AG and it is denoted by . In this paper, we gain some preferred results and limits that referring to the full domatic number on anti fuzzy graph.

Domination and Independence in Fuzzy Semigraphs

Objectives: To identify dominating and independent sets in fuzzy semigraphs by developing suitable methods. To examine dominating and independent sets in fuzzy semigraphs for their characteristics and behaviors. The goal is to examine the use of the concepts of domination and independence in the fields of analysis of social networks, security of networks, and allocation of resources. Methods: This study introduces the concept of strong arc fuzzy semigraph and explores various parameters such as dominating and independent sets. The Minimum Adjacent Dominating Number (ad-number) \gamma_a(\operatorname{𝓖}) and Maximum Adjacent Dominating Number (ad-number) \Gamma_a(\operatorname{𝓖}) , Minimum End Node Adjacent Dominating Number (ead-number) \gamma_{ea}(\operatorname{𝓖}) and Maximum End Node Adjacent Dominating Number (ead-number) \Gamma_{ae}(\operatorname{𝓖}) as well as Minimum Consecutive Adjacent Dominating Number (cad-number) \gamma_{ca}(\operatorname{𝓖}) and Maximum Consecutive Adjacent Dominating Number (cad-number) \Gamma_{ca}(\operatorname{𝓖}) . These parameters represent the respective minimum and maximum cardinalities taken over all minimal dominating sets of nodes of \operatorname{𝓖} . Additionally, it explores the e-independence dominating number i_e and e-independence number \beta_e(\operatorname{𝓖}) , strongly independence dominating number i_s and strongly independence number \beta_s(\operatorname{𝓖}) and consecutive adjacent independence dominating number i_{ca} and consecutive adjacent independence number \beta_{ca}(\operatorname{𝓖}) . These parameters represent the respective minimum and maximum cardinalities derived from all maximal independent sets of nodes in \operatorname{𝓖} . Findings: The study develops a thorough correlation between fuzzy dominant sets and independent sets in the fuzzy semigraph \operatorname{𝓖} . The statement illustrates the way in which the introduced factors interact and impact the structure and behaviour of the fuzzy semigraph. Novelty: The introduction of strong arc fuzzy semigraphs is a novel notion in the examination of fuzzy graphs. The study involves the creation and examination of novel parameters associated with dominant and independent sets in fuzzy semigraphs. This study aims to establish novel theoretical linkages between fuzzy domination and independent sets, therefore expanding our knowledge of the structural features of fuzzy semigraphs. Keywords: Domination in fuzzy semigraphs, Independent in fuzzy semigraphs, Adjacent-domination, ea-domination, ca-domination, e-independent, s-independent, ca-independent

The Maximum Zero-Sum Partition problem

We study the Maximum Zero-Sum Partition problem (or MZSP), defined as follows: given a multiset S={a1,a2,…,an} of integers ai∈Z⁎ (where Z⁎ denotes the set of non-zero integers) such that ∑i=1nai=0, find a maximum cardinality partition {S1,S2,…,Sk} of S such that, for every 1≤i≤k, ∑aj∈Siaj=0. Solving MZSP is useful in genomics for computing evolutionary distances between pairs of species. Our contributions are a series of algorithmic results concerning MZSP, in terms of complexity, (in)approximability, with a particular focus on the fixed-parameter tractability of MZSP with respect to either (i) the size k of the solution, (ii) the number of negative (resp. positive) values in S and (iii) the largest integer in S.

Finding conserved low‐diameter subgraphs in social and biological networks

AbstractThe analysis of social and biological networks often involves modeling clusters of interest as cliques or their graph‐theoretic generalizations. The ‐club model, which relaxes the requirement of pairwise adjacency in a clique to length‐bounded paths inside the cluster, has been used to model cohesive subgroups in social networks and functional modules or complexes in biological networks. However, if the graphs are time‐varying, or if they change under different conditions, we may be interested in clusters that preserve their property over time or under changes in conditions. To model such clusters that are conserved in a collection of graphs, we consider a cross‐graph ‐club model, a subset of nodes that forms a ‐club in every graph in the collection. In this article, we consider the canonical optimization problem of finding a cross‐graph ‐club of maximum cardinality in a graph collection. We develop integer programming approaches to solve this problem. Specifically, we introduce strengthened formulations, valid inequalities, and branch‐and‐cut algorithms based on delayed constraint generation. The results of our computational study indicate the significant benefits of using the approaches we introduce.

On spectral extrema of graphs with given order and generalized 4-independence number

Characterizing the graph having the maximum or minimum spectral radius in a given class of graphs is a classical problem in spectral extremal graph theory, originally proposed by Brualdi and Solheid. Given a graph G, a vertex subset S is called a maximum generalized 4-independent set of G if the induced subgraph G[S] dose not contain a 4-tree as its subgraph, and the subset S has maximum cardinality. The cardinality of a maximum generalized 4-independent set is called the generalized 4-independence number of G. In this paper, we firstly determine the connected graph (resp. bipartite graph, tree) having the largest spectral radius over all connected graphs (resp. bipartite graphs, trees) with fixed order and generalized 4-independence number, in addition, we establish a lower bound on the generalized 4-independence number of a tree with fixed order. Secondly, we describe the structure of all the n-vertex graphs having the minimum spectral radius with generalized 4-independence number ψ, where ψ⩾⌈3n/4⌉. Finally, we identify all the connected n-vertex graphs with generalized 4-independence number ψ∈{3,⌈3n/4⌉,n−1,n−2} having the minimum spectral radius.

On Minimal Geodetic Hop Domination in Graphs

Let $G$ be a nontrivial connected graph with vertex set $V(G)$. A set $S \subseteq V(G)$ is a geodetic hop dominating set of $G$ if the following two conditions hold for each $x\in V(G)\setminus S$: $(1)$ $x$ lies in some $u$-$v$ geodesic in $G$ with $u,v\in S$, and $(2)$ $x$ is of distance $2$ from a vertex in $S$. The minimum cardinality $\gamma_{hg}(G)$ of a geodetic hop dominating set of $G$ is the geodetic hop domination number of $G$. A geodetic hop dominating set $S$ is a minimal geodetic hop dominating set if $S$ does not contain a proper subset that is itself geodetic hop dominating set. The maximum cardinality of a minimal geodetic hop dominating set in $G$ is the \emph{upper geodetic hop domination number} of $G$, and is denoted by $\gamma^{+}_{hg}(G)$. This paper initiates the study of the minimal geodetic hop dominating set and the corresponding upper geodetic hop domination number of nontrivial connected graphs. Interestingly, every pair of positive integers $a$ and $b$ with $2\le a\le b$ is realizable as the geodetic domination number and the upper geodetic hop domination number, respectively, of some graph. Furthermore, this paper investigates the concept in the join, corona and lexicographic product of graphs.

Total coalitions in graphs

We define a total coalition in a graph G as a pair of disjoint subsets A 1,A 2 ⊆ A that satisfy the following conditions: (a) neither A 1 nor A 2 constitutes a total dominating set of G, and (b) A 1 ∪ A 2 constitutes a total dominating set of G. A total coalition partition of a graph G is a partition ϒ = {A 1,A 2, . ,Ak } of its vertex set such that no subset of ϒ acts as a total dominating set of G, but for every set Ai ∈ ϒ, there exists a set Aj ∈ ϒ such that Ai and Aj combine to form a total coalition. We define the total coalition number of G as the maximum cardinality of a total coalition partition of G, and we denote it by Ct (G). The purpose of this paper is to begin an investigation into the characteristics of total coalition in graphs.

Benchmarking quantum annealing with maximum cardinality matching problems

We benchmark Quantum Annealing (QA) vs. Simulated Annealing (SA) with a focus on the impact of the embedding of problems onto the different topologies of the D-Wave quantum annealers. The series of problems we study are especially designed instances of the maximum cardinality matching problem that are easy to solve classically but difficult for SA and, as found experimentally, not easy for QA either. In addition to using several D-Wave processors, we simulate the QA process by numerically solving the time-dependent Schrödinger equation. We find that the embedded problems can be significantly more difficult than the unembedded problems, and some parameters, such as the chain strength, can be very impactful for finding the optimal solution. Thus, finding a good embedding and optimal parameter values can improve the results considerably. Interestingly, we find that although SA succeeds for the unembedded problems, the SA results obtained for the embedded version scale quite poorly in comparison with what we can achieve on the D-Wave quantum annealers.

Bounds on the Maximum Cardinality of Indel and Substitution Correcting Codes

Bounds on the Maximum Cardinality of Indel and Substitution Correcting Codes

Shared-Memory Parallel Edmonds Blossom Algorithm for Maximum Cardinality Matching in General Graphs.

The Edmonds Blossom algorithm is implemented here using depth-first search, which is intrinsically serial. By streamlining the code, our serial implementation is consistently three to five times faster than the previously fastest general graph matching code. By extracting parallelism across iterations of the algorithm, with coarse-grain locking, we are able to further reduce the run time on random regular graphs four-fold and obtain a two-fold reduction of run time on real-world graphs with similar topology. Solving very sparse graphs (average degree less than four) exhibiting community structure with eight threads led to a slow down of three-fold, but this slow down is replaced by marginal speed up once the average degree is greater than four. We conclude that our parallel coarse-grain locking implementation performs well when extracting parallelism from this augmenting-path-based algorithm and may work well for similar algorithms.

Parallel Maximum Cardinality Matching for General Graphs on GPUs.

The matching problem formulated as Maximum Cardinality Matching in General Graphs (MCMGG) finds the largest matching on graphs without restrictions. The Micali-Vazirani algorithm has the best asymptotic complexity for solving MCMGG when the graphs are sparse. Parallelizing matching in general graphs on the GPU is difficult for multiple reasons. First, the augmenting path procedure is highly recursive, and NVIDIA GPUs use registers to store kernel arguments, which eventually spill into cached device memory, with a performance penalty. Second, extracting parallelism from the matching process requires partitioning the graph to avoid any overlapping augmenting paths. We propose an implementation of the Micali-Vazirani algorithm which identifies bridge edges using thread-parallel breadth-first search, followed by block-parallel path augmentation and blossom contraction. Augmenting path and Union-find methods were implemented as stack-based iterative methods, with a stack allocated in shared memory. Our experimentation shows that compared to the serial implementation, our approach results in up to 15-fold speed-up for very sparse regular graphs, up to 5-fold slowdown for denser regular graphs, and finally a 50-fold slowdown for power-law distributed Kronecker graphs. This implementation has been open-sourced for further research on developing combinatorial graph algorithms on GPUs.

A Theory of Alternating Paths and Blossoms from the Perspective of Minimum Length

The Micali–Vazirani (MV) algorithm for finding a maximum cardinality matching in general graphs, which was published in 1980, remains to this day the most efficient known algorithm for the problem. The current paper gives the first complete and correct proof of this algorithm. The MV algorithm resorts to finding minimum-length augmenting paths. However, such paths fail to satisfy an elementary property, called breadth first search honesty in this paper. In the absence of this property, an exponential time algorithm appears to be called for—just for finding one such path. On the other hand, the MV algorithm accomplishes this and additional tasks in linear time. The saving grace is the various “footholds” offered by the underlying structure, which the algorithm uses in order to perform its key tasks efficiently. The theory expounded in this paper elucidates this rich structure and yields a proof of correctness of the algorithm. It may also be of independent interest as a set of well-knit graph-theoretic facts. Funding: This work was supported in part by the National Science Foundation [Grant CCF-2230414].

Statistical methods to control for confounders in rare disease settings that use external control

ABSTRACT In the drug development for rare disease, the number of treated subjects in the clinical trial is often very small, whereas the number of external controls can be relatively large. There is no clear guidance on choosing an appropriate statistical method to control baseline confounding in this situation. To fill this gap, we conduct extensive simulations to evaluate the performance of commonly used matching and weighting methods as well as the more recently developed targeted maximum likelihood estimation (TMLE) and cardinality matching in small sample settings, mimicking the motivating data from a pediatric rare disease. Among the methods examined, the performance of coarsened exact matching (CEM) and TMLE are relatively robust under various model specifications. CEM is only feasible when the number of controls far exceeds the number of treated, whereas TMLE has better performance with less extreme treatment allocation ratios. Our simulations suggest bootstrap is useful for variance estimation in small samples after matching.

Gated independence in graphs

If G=(V,E) is a (finite and simple) graph, we call an independent set X a gated independent set in G if for each x∈X, there exists a neighbor y of x such that (X∖{x})∪{y} is an independent set in G. We define the gated independence numbergi(G) of G to be the maximum cardinality of a gated independent set in G. We demonstrate that the gated independence number is closely related to both matching and domination parameters of graphs. We prove that the inequalities im(G)⩽gi(G)⩽mur(G) hold for every graph G, where im(G) and mur(G) denote the induced and uniquely restricted matching numbers of G. On the other hand, we show that γi(G)⩽gi(G) and γpr(G)⩽2gi(G) for every graph G without any isolated vertex, where γi(G) and γpr(G) denote the independence and paired domination numbers. Furthermore, we provide bounds on the gated independence number involving the order, size and maximum degree. In particular, we prove that gi(G)⩾n5 for every n-vertex subcubic graph G without any isolated vertex or any component isomorphic to K3,3, and gi(B)⩽3n8 for every n-vertex connected cubic bipartite graph B.