Vertex Cover Research Articles

On minimal coverings and pairwise generation of some primitive groups of wreath product type

The covering number of a finite group [Formula: see text], denoted [Formula: see text], is the smallest positive integer [Formula: see text] such that [Formula: see text] is a union of [Formula: see text] proper subgroups. We calculate [Formula: see text] for a family of primitive groups [Formula: see text] with a unique minimal normal subgroup [Formula: see text], isomorphic to [Formula: see text] with [Formula: see text] divisible by [Formula: see text] and [Formula: see text] cyclic. This is a generalization of a result of Swartz concerning the symmetric groups. We also prove an asymptotic result concerning pairwise generation.

Matroid-constrained vertex cover

In this paper, we introduce the Matroid-Constrained Vertex Cover problem: given a graph with weights on the edges and a matroid imposed on the vertices, our problem is to choose a subset of vertices that is independent in the matroid, with the objective of maximizing the total weight of covered edges. This problem is a generalization of the much studied maxk-vertex cover problem, in which the matroid is the simple uniform matroid, and it is also a special case of the problem of maximizing a monotone submodular function under a matroid constraint.In the first part of this work, we give a Fixed-Parameter Tractable Approximation Scheme (FPT-AS) when the given matroid is a partition matroid, a laminar matroid, or a transversal matroid. Precisely, if k is the rank of the matroid, we obtain (1−ε) approximation using (1/ε)O(k)nO(1) time for partition and laminar matroids and using (1/ε+k)O(k)nO(1) time for transversal matroids. This extends a result of Manurangsi for uniform matroids [33]. We also show that these ideas can be applied in the context of (single-pass) streaming algorithms. Besides, our FPT-AS introduces a new technique based on matroid union, which may be of independent interest in extremal combinatorics.In the second part, we consider general matroids. We propose a simple local search algorithm that guarantees 2/3≈0.66 approximation. For the more general problem where two matroids are imposed on the vertices and a feasible solution must be a common independent set, we show that a local search algorithm gives a 2/3⋅(1−1/(p+1)) approximation in nO(p) time, for any integer p. We also provide some evidence to show that with the constraint of one or two matroids, the approximation ratio of 2/3 is likely the best possible, using the currently known techniques of local search.

Solving independent set problems with photonic quantum circuits

An independent set (IS) is a set of vertices in a graph such that no edge connects any two vertices. In adiabatic quantum computation [E. Farhi, et al., Science 292, 472-475 (2001); A. Das, B. K. Chakrabarti, Rev. Mod. Phys. 80, 1061-1081 (2008)], a given graph G(V, E) can be naturally mapped onto a many-body Hamiltonian [Formula: see text], with edges [Formula: see text] being the two-body interactions between adjacent vertices [Formula: see text]. Thus, solving the IS problem is equivalent to finding all the computational basis ground states of [Formula: see text]. Very recently, non-Abelian adiabatic mixing (NAAM) has been proposed to address this task, exploiting an emergent non-Abelian gauge symmetry of [Formula: see text] [B. Wu, H. Yu, F. Wilczek, Phys. Rev. A 101, 012318 (2020)]. Here, we solve a representative IS problem [Formula: see text] by simulating the NAAM digitally using a linear optical quantum network, consisting of three C-Phase gates, four deterministic two-qubit gate arrays (DGA), and ten single rotation gates. The maximum IS has been successfully identified with sufficient Trotterization steps and a carefully chosen evolution path. Remarkably, we find IS with a total probability of 0.875(16), among which the nontrivial ones have a considerable weight of about 31.4%. Our experiment demonstrates the potential advantage of NAAM for solving IS-equivalent problems.

Conservation agriculture assists smallholder farmers and their agroecosystem in the Democratic Republic of the Congo

Conservation agriculture, a sustainable farming practice combining no or minimum soil disturbance, crop diversification, and soil cover, can benefit humans and the biophysical environment. However, less than half of Sub-Saharan African countries implement conservation agriculture, and the majority of smallholder farmers are using unsustainable practices. Research and field trials on conservation agriculture adaptation to soil and agroecological conditions are vital for increasing adoption of conservation agriculture in Sub-Saharan Africa. The objectives of this study were to (i) evaluate the change in soil chemical properties and soil biology from implementation of conservation agriculture, (ii) assess cowpea (Vigna unguiculata L. Walp.) yield and variability under conservation agriculture compared to the conventional farming practice, and (iii) evaluate differences in soil chemical properties, soil biology, and crop yields under conservation agriculture in savannah and forest ecosystems compared to conventional agriculture in smallholder farming in the province of Maniema in the Democratic Republic of the Congo. Soils under conservation agriculture had 6.5 times more (p < 0.001) earthworms, 23% and 10% greater (p < 0.05) concentrations of soil phosphorus and potassium, respectively, and 100% greater (p < 0.001) cowpea yield than soils under conventional agriculture. Measured across all sites, conservation agriculture and conventional cowpea yields had coefficients of variation of 25% and 16%, respectively. This study showed for the first time that conservation agriculture implementation contributes to soil health and food security associated with smallholder farming in the Democratic Republic of the Congo.

OPTIMIST: Lightweight and Transparent IDS With Optimum Placement Strategy to Mitigate Mixed-Rate DDoS Attacks in IoT Networks

Distributed denial of service (DDoS) attacks are widespread for Internet of things (IoT) systems that aim to disrupt the availability of a system completely (high-rate DDoS) or partially (low-rate DDoS). Design and placement of Intrusion Detection Systems (IDS) for DDoS attacks on IoT systems are challenging due to the low power and lossy nature of networks. Existing IDSs are designed to handle either high-rate or low-rate DDoS but cannot handle both with good accuracy. Existing IDS placement techniques are mostly non-transparent, making malicious nodes aware of the presence of IDS nodes. Most of the IDS placement strategies are non-optimal, making them energy inefficient. Accordingly, this work proposes a transparent, optimally placed, distributed IDS solution, namely OPTIMIST, which can handle both high-rate and low-rate DDoS attacks with good accuracy. The placement problem is formulated as the weighted minimum vertex cover problem of a K-uniform hypergraph and solved with an approximation algorithm. The IDS module is based on a LSTM model where a novel offline training method for LSTM is proposed using WGAN-generated artificial flows. Extensive experimentation on simulation and testbed shows that the OPTIMIST can best achieve the balance between DDoS detection and energy overhead.

Near-optimal distributed dominating set in bounded arboricity graphs

We describe a simple deterministic O(ε-1logΔ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O( \\varepsilon ^{-1} \\log \\Delta )$$\\end{document} round distributed algorithm for (2α+1)(1+ε)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(2\\alpha +1)(1 + \\varepsilon )$$\\end{document} approximation of minimum weighted dominating set on graphs with arboricity at most α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}. Here Δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta $$\\end{document} denotes the maximum degree. We also show a lower bound proving that this round complexity is nearly optimal even for the unweighted case, via a reduction from the celebrated KMW lower bound on distributed vertex cover approximation (Kuhn et al. in JACM 63:116, 2016). Our algorithm improves on all the previous results (that work only for unweighted graphs) including a randomized O(α2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\\alpha ^2)$$\\end{document} approximation in O(logn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\\log n)$$\\end{document} rounds (Lenzen et al. in International symposium on distributed computing, Springer, 2010), a deterministic O(αlogΔ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\\alpha \\log \\Delta )$$\\end{document} approximation in O(logΔ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\\log \\Delta )$$\\end{document} rounds (Lenzen et al. in international symposium on distributed computing, Springer, 2010), a deterministic O(α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\\alpha )$$\\end{document} approximation in O(log2Δ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\\log ^2 \\Delta )$$\\end{document} rounds (implicit in Bansal et al. in Inform Process Lett 122:21–24, 2017; Proceeding 17th symposium on discrete algorithms (SODA), 2006), and a randomized O(α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\\alpha )$$\\end{document} approximation in O(αlogn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\\alpha \\log n)$$\\end{document} rounds (Morgan et al. in 35th International symposiumon distributed computing, 2021). We also provide a randomized O(αlogΔ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\\alpha \\log \\Delta )$$\\end{document} round distributed algorithm that sharpens the approximation factor to α(1+o(1))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha (1+o(1))$$\\end{document}. If each node is restricted to do polynomial-time computations, our approximation factor is tight in the first order as it is NP-hard to achieve α-1-ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha - 1 - \\varepsilon $$\\end{document} approximation (Bansal et al. in Inform Process Lett 122:21-24, 2017).

A reduction from an LWE problem to maximum independent set problems

The learning with errors (LWE) problem is a problem derived from machine learning that is believed to be intractable for quantum computers. This paper proposes a method that can reduce an LWE problem to a set of maximum independent set (MIS) problems, which are graph problems that are suitable for a quantum annealing machine to solve. The reduction algorithm can reduce an n-dimensional LWE problem to several small MIS problems with at most 2^{O(sqrt{n})} nodes when the lattice-reduction algorithm used in the LWE-reduction method successfully finds short vectors. The algorithm is useful for solving LWE problems in a quantum-classical hybrid manner by using an existing quantum algorithm to solve the MIS problems. For example, the smallest LWE challenge problem is reduced to MIS problems with about 40,000 vertices. This result means that the smallest LWE challenge problem will be within the scope of a real quantum computer in the near future.

A lightweight semi-centralized strategy for the massive parallelization of branching algorithms

Several NP-hard problems are solved exactly using exponential-time branching strategies, whether it be branch-and-bound algorithms, or bounded search trees in fixed-parameter algorithms. The number of tractable instances that can be handled by sequential algorithms is usually small, whereas massive parallelization has been shown to significantly increase the space of instances that can be solved exactly. However, previous centralized approaches require too much communication to be efficient, whereas decentralized approaches are more efficient but have difficulty keeping track of the global state of the exploration.In this work, we propose to revisit the centralized paradigm while avoiding previous bottlenecks. In our strategy, the center has lightweight responsibilities, requires only a few bits for every communication, but is still able to keep track of the progress of every worker. In particular, the center never holds any task but is able to guarantee that a process with no work always receives the highest priority task globally.Our strategy was implemented in a generic C++ library called GemPBA, which allows a programmer to convert a sequential branching algorithm into a parallel version by changing only a few lines of code. An experimental case study on the vertex cover problem demonstrates that some of the toughest instances from the DIMACS challenge graphs that would take months to solve sequentially can be handled within two hours with our approach.

Novel modified ANFIS based fuzzy logic model for performance prediction of FRCM-to-concrete bond strength

Construction repairs have used fiber-reinforced cement mortar (FRCM). Concrete and FRCM bond strength usually outweigh mechanical criteria. Nevertheless, testing complex bonds like the FRCM and concrete bond takes time, money, and errors. This study employed fuzzy logic (FL) based on the adaptive neuro-fuzzy inference system (ANFIS) to simplify and reliably estimate the FRCM-to-concrete bond strength (CBS) by modeling complicated and non-linear systems computationally efficiently. The models take six inputs: concrete splice length, stirrup cross-sectional area to spacing, longitudinal tension bar area to effective cross section, compressive strength, relative rib area, and minimum concrete cover. The model outputs concrete steel bar bond strength. The FLANFIS model predicts FRCM-to-CBS using the tensile testing results of the 10 specimens (5 concrete and 5 FRCM). Data instruct the model and measure its precision. This article defines the ANFIS-based FRCM-to-CBS. This research will employ 5 concrete specimens and 5 FRCM specimens, totaling 0.05 m^3 of concrete and FRCM mix. 0.25 kg of adhesive bonds the sample. Grip, control, and data gathering systems are employed with a 1 kN tensile testing equipment. This research comprises preparing concrete and FRCM specimens, bonding with adhesive, and tensile testing. The FL-ANFIS model predicts FRCM-to-CBS with a high coefficient of determination (R2) of 0.995 and a strong correlation coefficient (r) of 0.982 in training. The pattern predicted accurately with RMSE of 0.264 and MAE of 0.196. This paper shows that FL-ANFIS can predict steel bar bond strength in concrete quickly and accurately. The pattern reduces waste, design costs, and time.

Independence and domination in divisor graph and mod-difference graphs

We initiate the study of domination and inverse domination in labeled graphs. In this paper, we determined the cardinality of maximal independent and minimum variant dominating (total dominating/independent dominating/co-independent dominating) sets and their inverse in divisor graph and in two new labeling definitions called 0-mod-difference and 1-mod-difference graphs.

On the number of maximal independent sets: From Moon–Moser to Hujter–Tuza

AbstractWe connect two classical results in extremal graph theory concerning the number of maximal independent sets. The maximum number of maximal independent sets in an ‐vertex graph was determined by Miller and Muller and independently by Moon and Moser. The maximum number of maximal independent sets in an ‐vertex triangle‐free graph was determined by Hujter and Tuza. We give a common generalization of these results by determining the maximum number of maximal independent sets in an ‐vertex graph containing no induced triangle matching of size . This also improves a stability result of Kahn and Park on . Our second result is a new (short) proof of a second stability result of Kahn and Park on the maximum number of maximal independent sets in ‐vertex triangle‐free graphs containing no induced matching of size .

Spanning k-Ended Tree in 2-Connected Graph

Win proved a very famous conclusion that states the graph G with connectivity κ(G), independence number α(G) and α(G)≤κ(G)+k−1(k≥2) contains a spanning k-ended tree. This means that there exists a spanning tree with at most k leaves. In this paper, we strengthen the Win theorem to the following: Let G be a simple 2-connected graph such that |V(G)|≥2κ(G)+k, α(G)≤κ(G)+k(k≥2) and the number of maximum independent sets of cardinality κ+k is at most n−2κ−k+1. Then, either G contains a spanning k-ended tree or a subgraph of Kκ∨((k+κ−1)K1∪Kn−2κ−k+1).

Communities Detection in Multiplex Networks Using Optimization: Study Case—Employment in Mexico during the COVID-19 Pandemic

The detection of communities in complex networks offers important information about the structure of the network as well as its dynamics. However, it is not an easy problem to solve. This work presents a methodology based of the robust coloring problem (RCP) and the vertex cover problem (VCP) to find communities in multiplex networks. For this, we consider the RCP idea of having a partial detection based onf the similarity of connected and unconnected nodes. On the other hand, with the idea of the VCP, we manage to minimize the number of groups, which allows us to identify the communities well. To apply this methodology, we present the dynamic characterization of job loss, change, and acquisition behavior for the Mexican population before and during the COVID-19 pandemic modeled as a 4- layer multiplex network. The results obtained when applied to test and study case networks show that this methodology can classify elements with similar characteristics and can find their communities. Therefore, our proposed methodology can be used as a new mechanism to identify communities, regardless of the topology or whether it is a monoplex or multiplex network.

Very well-covered graphs with the Erdős–Ko–Rado property

A family of independent $r$-sets of a graph $G$ is an $r$-star if every set in the family contains some fixed vertex $v$. A graph is $r$-EKR if the maximum size of an intersecting family of independent $r$-sets is the size of an $r$-star. Holroyd and Talbot conjecture that a graph is $r$-EKR as long as $1\leq r\leq\frac{\mu(G)}{2}$, where $\mu(G)$ is the minimum size of a maximal independent set. It is suspected that the smallest counterexample to this conjecture is a well-covered graph. Here we consider the class of very well-covered graphs $G^*$ obtained by appending a single pendant edge to each vertex of $G$. We prove that the pendant complete graph $K_n^*$ is $r$-EKR when $n \geq 2r$ and strictly so when $n>2r$. Pendant path graphs $P_n^*$ are also explored and the vertex whose $r$-star is of maximum size is determined.

From symmetry to asymmetry: Generalizing TSP approximations by parametrization

We generalize the tree doubling and Christofides algorithm to parameterized approximations for ATSP (constant factor approximations that invest more runtime with respect to a chosen parameter). The parameters we consider are upper bounded by the number of asymmetric distances, which yields algorithms to efficiently compute good approximations for moderately asymmetric TSP instances. As generalization of the Christofides algorithm, we derive a parameterized 2.5-approximation, with the size of a vertex cover for the subgraph induced by the edges with asymmetric distances as parameter. Our generalization of tree doubling gives a parameterized 3-approximation, where the parameter is the minimum number of asymmetric distances in a minimum spanning arborescence. Further, we combine these with a notion of symmetry relaxation which allows to trade approximation guarantee for runtime. Since the parameters we consider are theoretically incomparable, we present experimental results showing that generalized tree doubling frequently outperforms generalized Christofides with respect to parameter size.

The dominant edge metric dimension of graphs

For an ordered subset S = { v 1 , …, v k } of vertices in a connected graph G and an edge e ′ of G , the edge metric S -representation of e ′= a b is the vector r G e ( e ′| S )=( d G ( e ′, v 1 ),…, d G ( e ′, v k )) , where d G ( e ′, v i )=min{ d G ( a , v i ), d G ( b , v i )} . A dominant edge metric generator for G is a vertex cover S of G such that the edges of G have pairwise different edge metric S -representations. A dominant edge metric generator of smallest size of G is called a dominant edge metric basis for G . The size of a dominant edge metric basis of G is denoted by D d i m e ( G ) and is called the dominant edge metric dimension. In this paper, the concept of dominant edge metric dimension (DEMD for short) is introduced and its basic properties are studied. Moreover, NP-hardness of computing DEMD of connected graphs is proved. Furthermore, this invariant is investigated under some graph operations at the end of the paper.

A 5k-vertex kernel for 3-path vertex cover

A 3-path vertex cover in a graph is a vertex subset C such that every path of three vertices contains at least one vertex from C. The parameterized 3-path vertex cover problem asks whether a graph has a 3-path vertex cover of size at most k. We give a kernel of 5k vertices for this problem, improving the previously known bounds.

Optimal Scale-Free Small-World Graphs with Minimum Scaling of Cover Time

The cover time of random walks on a graph has found wide practical applications in different fields of computer science, such as crawling and searching on the World Wide Web and query processing in sensor networks, with the application effects dependent on the behavior of the cover time: the smaller the cover time, the better the application performance. It was proved that over all graphs withNnodes, complete graphs have the minimum cover timeNlogN. However, complete graphs cannot mimic real-world networks with small average degree and scale-free small-world properties, for which the cover time has not been examined carefully, and its behavior is still not well understood. In this article, we first experimentally evaluate the cover time for various real-world networks with scale-free small-world properties, which scales asNlogN. To better understand the behavior of the cover time for real-world networks, we then study the cover time of three scale-free small-world model networks by using the connection between cover time and resistance diameter. For all the three networks, their cover time also behaves asNlogN. This work indicates that sparse networks with scale-free and small-world topology are favorable architectures with optimal scaling of cover time. Our results deepen understanding the behavior of cover time in real-world networks with scale-free small-world structure, and have potential implications in the design of efficient algorithms related to cover time.

Probability weighting effect in vertex cover of networks via prospect-theoretic learning

Game-theoretic learning methods for the vertex cover problem have been investigated in this paper. In the traditional game theory, the establishment of the game model is based on the complete objectivity of the players, and the existing game models describe the vertex cover problem mainly along this path. In contrast, this paper considers the impact of players' subjectivity on decision-making results. First, we present a covering game model, where the utility function of the player is established under the prospect theory. Then, by presenting a rounding function, the states of all vertices under Nash equilibrium satisfies vertex cover state of a general network. After then, we present a fictitious play distributed algorithm, which can guarantee that the states of all vertices converge a Nash equilibrium. Finally, the simulation results are presented to assess the impact of players' subjectivity on the overall cover results of networks.

Some bounds on the size of maximum G-free sets in graphs

For a given graph [Formula: see text], the independence number [Formula: see text] of [Formula: see text] is the size of the maximum independent set of [Formula: see text]. Finding the maximum independent set in a graph is NP-hard. Another version of the independence number is defined as the size of the maximum-induced forest of [Formula: see text], and called the forest number of [Formula: see text], and denoted by [Formula: see text]. Finding [Formula: see text] is also an NP-hard problem. Suppose that [Formula: see text] is a graph, and [Formula: see text] is a family of graphs, a graph [Formula: see text] has a [Formula: see text]-free [Formula: see text]-coloring if there exists a decomposition of [Formula: see text] into sets [Formula: see text], [Formula: see text], so that [Formula: see text] for each [Formula: see text], and each [Formula: see text]. [Formula: see text] is [Formula: see text]-free, if the subgraph of [Formula: see text] induced by [Formula: see text] is [Formula: see text]-free, i.e., it contains no copy of [Formula: see text]. Finding a maximum subset [Formula: see text] of [Formula: see text], such that [Formula: see text] is a [Formula: see text]-free graph is a very hard problem as well. In this paper, we study the generalized version of the independence number of a graph. Also, we give some bounds about the size of the maximum [Formula: see text]-free subset of graphs as another purpose of this paper.