Vertex Cover Research Articles

Component Intersection Graph of Finite Dimensional Vector Spaces

In this paper, we introduce a graph structure, called component intersection graph, on a finite dimensional vector space V. The connectivity, diameter, maximal independent sets, clique number, chromatic number of component intersection graph have been studied.

On Critical Node Problems with Vulnerable Vertices

A vertex pair in an undirected graph is called \emph{connected} ifthe two vertices are connected by a path. In the NP-hard\textsc{Critical Node Problem}~(CNP), the input is an undirectedgraph~$G$ with integers~$k$ and~$x$, and the question is whether onecan transform~$G$ by deleting at most~$k$ vertices into a graph whose totalnumber of connected vertex pairs is at most~$x$. In this work, weintroduce and study two NP-hard variants of CNP where a subset ofthe vertices is marked as \emph{vulnerable}, and we aim to obtain agraph with at most~$x$ connected vertex pairs containing at least one vulnerable vertex. In the first variant, which generalizes CNP,we may delete vulnerable and non-vulnerable vertices. Inthe second variant, we may only delete non-vulnerable vertices. We perform a parameterized complexity study of both problems. For example, we show that both problems are FPT with respect to~$k+x$. Furthermore, in the case of deletable vulnerable nodes, we provide a polynomial kernel for the parameter~$vc+k$, where~$vc$ is the vertex cover number. In the case of non-deletable vulnerable nodes, we prove NP-hardness even when there is only one vulnerable node.

A quadratic-order problem kernel for the traveling salesman problem parameterized by the vertex cover number

The NP-hard graphical traveling salesman problem (GTSP) is to find a closed walk of total minimum weight that visits each vertex in an undirected edge-weighted and not necessarily complete graph. We present a problem kernel with τ2+τ vertices for GTSP, where τ is the vertex cover number of the input graph. Any α-approximate solution for the problem kernel also gives an α-approximate solution for the original instance, for any α≥1.

An improved approximation algorithm for the <inline-formula><tex-math id="M1">$ k $</tex-math></inline-formula>-prize-collecting minimum power cover problem

An improved approximation algorithm for the <inline-formula><tex-math id="M1">$ k $</tex-math></inline-formula>-prize-collecting minimum power cover problem

Yet Another Self-stabilizing Minimum Vertex Cover of a Network with Stochastic Stability

Yet Another Self-stabilizing Minimum Vertex Cover of a Network with Stochastic Stability

Maximal independence and symmetry in crystal chemistry of natural tectosilicates.

Löwenstein's avoidance rule in aluminosilicates is reinterpreted on the basis of the fourth Pauling rule. It is shown that avoidance of Si-O-Si bridges may account for avoidance of Al-O-Al bridges. In view of this interpretation, it is proposed that the most favourable distributions of cations entering in substitution of silicon in the framework are associated to maximal independent sets of the respective 3-periodic nets. Among all possible solutions, only those with maximal symmetry are realized. The applicability of the concept is demonstrated for a few natural tectosilicates, which have been analysed through the prism of their labelled quotient graph.

On the packing number of $ 3 $-token graph of the path graph $ P_n $

<abstract><p>In 2018, J. M. Gómez et al. showed that the problem of finding the packing number $ \rho(F_2(P_n)) $ of the 2-token graph $ F_2(P_n) $ of the path $ P_n $ of length $ n\ge 2 $ is equivalent to determining the maximum size of a binary code $ S' $ of constant weight $ w = 2 $ that can correct a single adjacent transposition. By determining the exact value of $ \rho(F_2(P_n)) $, they proved a conjecture of Rob Pratt. In this paper, we study a related problem, which consists of determining the packing number $ \rho(F_3(P_n)) $ of the graph $ F_3(P_n) $. This problem corresponds to the Sloane's problem of finding the maximum size of $ S' $ of constant weight $ w = 3 $ that can correct a single adjacent transposition. Since the maximum packing set problem is computationally equivalent to the maximum independent set problem, which is an NP-hard problem, then no polynomial time algorithms are expected to be found. Nevertheless, we compute the exact value of $ \rho(F_3(P_n)) $ for $ n\leq 12 $, and we also present some algorithms that produce a lower bound for $ \rho(F_3(P_n)) $ with $ 13\leq n\leq 44 $. Finally, we establish an upper bound for $ \rho(F_3(P_n)) $ with $ n\geq 13 $.</p></abstract>

Proposição de variação do cobrimento nominal das armaduras baseada na classe de agressividade ambiental (CAA) de exposição de concretos

Abstract The physical and chemical protection of steel reinforcement afforded by the cover is a consequence of the quality of the concrete and its thickness, with a long-term impact on durability and service life. This study proposed variations in cover based on environmental aggressivity class (EAC) of exposure considering concretes of prescribed compressive strength classes (C20/C25/C30/C40). To this end, concrete mixtures were subjected to accelerated testing in salt spray chambers or carbonation. Accelerated testing was kept for periods long enough to reach each EAC cover level. Thus, the effect of each EAC, from mild to severe, was determined on the physical and mechanical characteristics of each mixture. It was determined that the required reinforcement cover varied linearly with compressive strength. This denoted the possibility of using more than one class of concrete and cover for each EAC. It should be noted that the difference in required cover became greater when using concrete with a strength class higher than C30. Still, it was noticed that the chloride ions attack s was more severe than carbonation and was, for most of the test cases, the determinant factor in minimum cover thickness to ensure the desired durability.

Kernelization for feedback vertex set via elimination distance to a forest

We study efficient preprocessing for the undirected Feedback Vertex Set problem, a fundamental problem in graph theory which asks for a minimum-sized vertex set whose removal yields an acyclic graph. More precisely, we aim to determine for which parameterizations this problem admits a polynomial kernel. While a characterization is known for the related Vertex Cover problem based on the recently introduced notion of bridge-depth, it remained an open problem whether this could be generalized to Feedback Vertex Set. The answer turns out to be negative; the existence of polynomial kernels for structural parameterizations for Feedback Vertex Set is governed by the elimination distance to a forest. Under the standard assumption NP⁄⊆coNP/poly, we prove that for any minor-closed graph class G, Feedback Vertex Set parameterized by the size of a modulator to G has a polynomial kernel if and only if G has bounded elimination distance to a forest. This captures and generalizes all existing kernels for structural parameterizations of the Feedback Vertex Set problem. [Display omitted]

Greedy maximal independent sets via local limits

AbstractThe random greedy algorithm for finding a maximal independent set in a graph constructs a maximal independent set by inspecting the graph's vertices in a random order, adding the current vertex to the independent set if it is not adjacent to any previously added vertex. In this paper, we present a general framework for computing the asymptotic density of the random greedy independent set for sequences of (possibly random) graphs by employing a notion of local convergence. We use this framework to give straightforward proofs for results on previously studied families of graphs, like paths and binomial random graphs, and to study new ones, like random trees and sparse random planar graphs. We conclude by analysing the random greedy algorithm more closely when the base graph is a tree.

The Variability of Snow Cover and Its Contribution to Water Resources in the Chinese Altai Mountains from 2000 to 2022

As one of the major water supply systems for inland rivers, especially in arid and semi-arid regions, snow cover strongly affects hydrological cycles. In this study, remote sensing datasets combined with in-situ observation data from a route survey of snow cover were used to investigate the changes in snow cover parameters on the Chinese Altai Mountains from 2000 to 2022, and the responses of snow cover to climate and hydrology were also discussed. The annual snow cover frequency (SCF), snow cover area, snow depth (SD), and snow density were 45.03%, 2.27 × 104 km2, 23.4 cm, and ~0.21 g·cm−3, respectively. The snow water equivalent ranged from 0.58 km3 to 1.49 km3, with an average of 1.12 km3. Higher and lower SCF were mainly distributed at high elevations and on both sides of the Irtysh river. The maximum and minimum snow cover parameters occurred in the Burqin River Basin and the Lhaster River Basin. In years with high SCF, abnormal westerly airflow was favorable for water vapor transport to the Chinese Altai Mountains, resulting in strong snowfall, and vice versa in years with low SCF. There were significant seasonal differences in the impact of temperature and precipitation on regional SCF changes. The snowmelt runoff ratios were 11.2%, 25.30%, 8.04%, 30.22%, and 11.56% in the Irtysh, Kayit, Haba, Kelan, and Burqin River Basins. Snow meltwater has made a significant contribution to the hydrology of the Chinese Altai Mountains.

Service Life Design of Concrete Structures Made of High-Volume Limestone Powder Concrete—Case of the Carbonation-Induced Corrosion

One of the paths to CO2 emissions reduction in the concrete industry is to use low-clinker cements, providing at the same time the performance of concrete that is adequate for application in concrete structures. This paper explores the impact of the clinker replacement with high amounts of limestone powder (21–70% in the powder phase) on concrete carbonation resistance. To quantify this impact, the empirical relationship between the carbonation resistance and the compressive strength of the high-volume limestone powder concrete (HVLPC) was established. For that purpose, the regression analysis was applied on the experimental results collected from the published research. The service life analysis based on the full probabilistic approach was performed using the fib Model Code 2010 prediction model and proposed empirical relationship. The first-order reliability method (FORM) was applied to solve the limit state function of reinforcement depassivation with a reliability index equal to 1.3. The obtained minimum concrete cover depths were 40–110% higher compared to those prescribed in the current European standard EN 1992-1-1:2004 for indicative strength classes. Based on the full probabilistic analysis, recommended cover depths are given for all carbonation exposure classes, commonly applied concrete strength classes, and service lives of 50 and 100 years.

Hardness and approximation of submodular minimum linear ordering problems

AbstractThe minimum linear ordering problem (MLOP) generalizes well-known combinatorial optimization problems such as minimum linear arrangement and minimum sum set cover. MLOP seeks to minimize an aggregated cost $$f(\cdot )$$ f ( · ) due to an ordering $$\sigma $$ σ of the items (say [n]), i.e., $$\min _{\sigma } \sum _{i\in [n]} f(E_{i,\sigma })$$ min σ ∑ i ∈ [ n ] f ( E i , σ ) , where $$E_{i,\sigma }$$ E i , σ is the set of items mapped by $$\sigma $$ σ to indices [i]. Despite an extensive literature on MLOP variants and approximations for these, it was unclear whether the graphic matroid MLOP was NP-hard. We settle this question through non-trivial reductions from mininimum latency vertex cover and minimum sum vertex cover problems. We further propose a new combinatorial algorithm for approximating monotone submodular MLOP, using the theory of principal partitions. This is in contrast to the rounding algorithm by Iwata et al. (in: APPROX, 2012), using Lovász extension of submodular functions. We show a $$(2-\frac{1+\ell _{f}}{1+|E|})$$ ( 2 - 1 + ℓ f 1 + | E | ) -approximation for monotone submodular MLOP where $$\ell _{f}=\frac{f(E)}{\max _{x\in E}f(\{x\})}$$ ℓ f = f ( E ) max x ∈ E f ( { x } ) satisfies $$1 \le \ell _f \le |E|$$ 1 ≤ ℓ f ≤ | E | . Our theory provides new approximation bounds for special cases of the problem, in particular a $$(2-\frac{1+r(E)}{1+|E|})$$ ( 2 - 1 + r ( E ) 1 + | E | ) -approximation for the matroid MLOP, where $$f = r$$ f = r is the rank function of a matroid. We further show that minimum latency vertex cover is $$\frac{4}{3}$$ 4 3 -approximable, by which we also lower bound the integrality gap of its natural LP relaxation, which might be of independent interest.

Unified bounds for the independence number of graphs

Abstract The Hoffman ratio bound, Lovász theta function, and Schrijver theta function are classical upper bounds for the independence number of graphs, which are useful in graph theory, extremal combinatorics, and information theory. By using generalized inverses and eigenvalues of graph matrices, we give bounds for independence sets and the independence number of graphs. Our bounds unify the Lovász theta function, Schrijver theta function, and Hoffman-type bounds, and we obtain the necessary and sufficient conditions of graphs attaining these bounds. Our work leads to some simple structural and spectral conditions for determining a maximum independent set, the independence number, the Shannon capacity, and the Lovász theta function of a graph.

S-Club Cluster Vertex Deletion on interval and well-partitioned chordal graphs

In this paper, we study the computational complexity of s-Club Cluster Vertex Deletion. Given a graph, s-Club Cluster Vertex Deletion (s-CVD) aims to delete the minimum number of vertices from the graph so that each connected component of the resulting graph has a diameter at most s. When s=1, the corresponding problem is popularly known as Cluster Vertex Deletion (CVD). We provide a faster algorithm for s-CVD on interval graphs. For each s≥1, we give an O(n(n+m))-time algorithm for s-CVD on interval graphs with n vertices and m edges. In the case of s=1, our algorithm is a slight improvement over the O(n3)-time algorithm of Cao et al. (2018), and for s≥2, it significantly improves the state-of-the-art running time On4.We also give a polynomial-time algorithm to solve CVD on well-partitioned chordal graphs, a graph class introduced by Ahn et al. (WG 2020) as a tool for narrowing down complexity gaps for problems that are hard on chordal graphs, and easy on split graphs. Our algorithm relies on a characterisation of the optimal solution and on solving polynomially many instances of the Weighted Bipartite Vertex Cover. This generalises a result of Cao et al. (2018) on split graphs. We also show that for any even integer s≥2, s-CVD is NP-hard on well-partitioned chordal graphs.

Maximum Bipartite Subgraphs of Geometric Intersection Graphs

We study the Maximum Bipartite Subgraph ([Formula: see text]) problem, which is defined as follows. Given a set [Formula: see text] of [Formula: see text] geometric objects in the plane, we want to compute a maximum-size subset [Formula: see text] such that the intersection graph of the objects in [Formula: see text] is bipartite. We first give an [Formula: see text]-time algorithm that computes an almost optimal solution for the problem on circular-arc graphs. We show that the [Formula: see text] problem is [Formula: see text]-hard on geometric graphs for which the maximum independent set is [Formula: see text]-hard (hence, it is [Formula: see text]-hard even on unit squares and unit disks). On the other hand, we give a [Formula: see text] for the problem on unit squares and unit disks. Moreover, we show fast approximation algorithms with small-constant factors for the problem on unit squares, unit disks, and unit-height axis parallel rectangles. Additionally, we prove that the Maximum Triangle-free Subgraph ([Formula: see text]) problem is NP-hard for axis-parallel rectangles. Here the objective is the same as that of the [Formula: see text] except the intersection graph induced by the set [Formula: see text] needs to be triangle-free only (instead of being bipartite).

Virtual Network Embedding Over Multi-Band Elastic Optical Network Based on Cross-Matching Mechanism and Hypergraph Theory

The commercialization of 5G and the explosive emergence of new applications stimulate the exponential growth of network traffic and diversification of services. It is promising to integrate multi-band elastic optical network (MBEON) and network virtualization for large volume traffic transmission and highly diverse services. However, performing virtual network embedding (VNE) for network virtualization over MBEON faces the challenge of severe inter-channel stimulated Raman scattering (ISRS) effect, which complicates the underlying physical layer effect in the substrate network. In this paper, we investigate the ISRS-aware VNE over MBEON, where a cross-matching mechanism is proposed for virtual node mapping (VNM) and a hypergraph is introduced for parallel virtual link mapping (VLM). A lightpath-level integer linear programming model is first formulated. To integrate the cost and availability of VLM, which significantly affect the performance of the VNE under the ISRS effect, into the VNM process, the “virtual node-substrate node” mapping pairs are specifically evaluated through the cross-matching mechanism. Moreover, to tackle the couplings among multiple lightpaths induced by the wide spectrum ISRS effect, hypergraphs are used to model the ISRS effect-aware quality of transmission (QoT) constraints among multiple lightpaths. A hypergraph maximal weight independent set heuristic is presented for lightpath selection, which guarantees the obedience of basic constraints and generates near-optimal solutions. Experimental results show that the proposed methods decrease blocking ratio by more than 30% compared with the benchmarks with similar computational complexity.

Observability and Detectability of Stochastic Labeled Graphs

In this paper, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">observable stochastic graphs and detetectable stochastic graphs</i> are, respectively, defined with the detailed implementation for the observability and detectability of stochastic discrete-time and discrete-state dynamic systems. More specifically, they are generally two classes of vertex-colored and edge-labeled graphs rendering a walking agent therein to determine his initial and current positions, respectively, in probability one by measuring the color sequence of his traversed vertices. In the part of analysis, we follow the aforementioned two definitions and establish the corresponding polynomial-time verifying algorithms. Notably, the implicit formulas are also established to calculate the observability and detectability probability for any pairwise vertex pair. In the synthesis part, the minimal number of colors dyeing the vertices to make any considered graph stochastically observable and detectable is investigated, respectively. Our results indicate that the minimal coloring problem is NP-hard for observability in the stochastic situations but is solvable in polynomial time for detectability in the deterministic cases. The observability and detectability of stochastic finite-valued systems, assembling with the finite-cardinal state spaces, are validated as a compelling application of these two types of directed graphs, whereas the minimal sensors placement problems subject to observability and detectability problems are accordingly interpreted by the minimum set cover algorithm.

Exact algorithms for restricted subset feedback vertex set in chordal and split graphs

The Restricted Subset Feedback Vertex Set problem (R-SFVS) takes a graph G=(V,E), a terminal set T⊆V, and an integer k as the input. The task is to determine whether there exists a subset S⊆V∖T of at most k vertices, after deleting which no terminal in T is contained in a cycle in the remaining graph. R-SFVS is NP-complete even when the input graph is restricted to split graphs. In this paper, we mainly show that R-SFVS in chordal and split graphs can be solved in O(1.1550|V|) time and exponential space or in O(1.1605|V|) time and polynomial space, significantly improving all previous results. As a by-product, we show that the Maximum Independent Set problem parameterized by the edge clique cover number is fixed-parameter tractable. Furthermore, by using a simple reduction from R-SFVS to Vertex Cover, we obtain an O⁎(1.2738k)-time parameterized algorithm and a tight O(k2)-kernel for R-SFVS in chordal and split graphs.

Heterogeneous decision-making dynamics of threshold-switching agents on complex networks.

In the classical two-player decision-making scenario, individuals may have different tendencies to take a certain action, given that there exists a sufficient number of neighbors adopting a particular option. This is ubiquitous in many real-life contexts including traffic congestion, crowd evacuation, and minimal vertex cover problem. Under best-response dynamics, we investigate the decision-making behaviors of heterogeneous agents on complex networks. Results of the networked games are twofold: for networks of uniform degree distribution (e.g., the lattice) and fraction of the strategy is of a linear function of the threshold setting. Moreover, the equilibrium analysis is provided and the relationship between the equilibrium dynamics and the change of the threshold value is given quantitatively. Next, if the games are played on networks with non-uniform degree distribution (e.g., random regular and scale-free networks), influence of the threshold-switching will be weakened. Robust experiments indicate that it is not the value of the average degree, but the degree distribution that influences how the strategy evolves affected by the threshold settings. Our result shows that the decision-making behaviors can be effectively manipulated by tuning the parameters in the utility function (i.e., thresholds) of some agents for more regular network structures.