Independence number and the number of maximum independent sets in pseudofractal scale-free web and Sierpiński gasket

03/02/2018 ∙ by Liren Shan, et al. ∙ FUDAN University 0

As a fundamental subject of theoretical computer science, the maximum independent set (MIS) problem not only is of purely theoretical interest, but also has found wide applications in various fields. However, for a general graph determining the size of a MIS is NP-hard, and exact computation of the number of all MISs is even more difficult. It is thus of significant interest to seek special graphs for which the MIS problem can be exactly solved. In this paper, we address the MIS problem in the pseudofractal scale-free web and the Sierpiński gasket, which have the same number of vertices and edges. For both graphs, we determine exactly the independence number and the number of all possible MISs. The independence number of the pseudofractal scale-free web is as twice as the one of the Sierpiński gasket. Moreover, the pseudofractal scale-free web has a unique MIS, while the number of MISs in the Sierpiński gasket grows exponentially with the number of vertices.

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1 Introduction

An independent set of a graph with vertex set is a subset of , such that each pair of vertices in is not adjacent in . A maximal independent set is an independent set that is not a subset of any other independent set. A largest maximal independent set is called a maximum independent set (MIS). In other words, a MIS is an independent set that has the largest size or cardinality. The cardinality of a MIS is referred to as the independence number of graph . A graph is called a unique independence graph if it has a unique MIS Hopkins and Staton (1985). The MIS problem has a close connection with many other fundamental graph problems Robson (1986); Berman and Fürer (1994); Halldórsson and Radhakrishnan (1997). For instance, the MIS problem in a graph is equivalent to the minimum vertex cover problem Karp (1972) in the same graph, as well as the maximum clique problem in its complement graph Pardalos and Xue (1994). In addition, the MIS problem is also closely related to graph coloring, maximum common induced subgraphs, and maximum common edge subgraphs Liu et al. (2015).

In addition to its intrinsic theoretical interest, the MIS problem has found important applications in a large variety of areas, such as coding theory Butenko et al. (2002), collusion detection in voting pools Araujo et al. (2011), scheduling in wireless networks Joo et al. (2016). For example, it was shown in Butenko et al. (2002) that the problem of finding the largest error correcting codes can be reduced to the MIS problem on a graph. In Araujo et al. (2011) the problem of collusion detection was framed as identifying maximum independent sets. Moreover, finding a maximal weighted independent set in a wireless network is connected with the problem of organizing the vertices of the network in a hierarchical way Basagni (2001). Finally, the MIS problem also has numerous applications in mining of graph data Liu et al. (2015); Chang et al. (2017).

In view of the theoretical and practical relevance, in the past decades the MIS problem has received much attention from different disciplines, e.g., theoretical computer science Murat and Paschos (2002); Xiao and Nagamochi (2013); Agnarsson et al. (2013); Hon et al. (2015); Lozin et al. (2015); Chuzhoy and Ene (2016) and discrete mathematics Čenek and Stewart (2003); Xiao and Nagamochi (2016); Mosca (2017). It is well-known that solving the MIS problem of a generic graph is computationally difficult. Finding a MIS of a graph is a classic NP-hard problem Robson (1986); Halldórsson and Radhakrishnan (1997), while enumerating all MISs in a graph is even #P-complete Valiant (1979a, b). Due to the hardness of the MIS problem, exact algorithms for finding a MIS in a general graph take exponential time Fomin and Kratsch (2010); Tarjan and Trojanowski (1977); Beame et al. (2007)

, which is infeasible for moderately sized graphs. For practical applications, many local or heuristic algorithms were proposed to solve the MIS problem for those massive and intractable graphs 

Andrade et al. (2012); Dahlum et al. (2016); Lamm et al. (2017).

Comprehensive empirical study Newman (2003) has unveiled that large real networks are typically scale-free Barabási and Albert (1999), with their vertex degree following a power-law distribution . This nontrivial heterogeneous structure has a strong effect on various topological and combinatorial aspects of a graph, such as average distances Chung and Lu (2002), maximum matchings Liu et al. (2011); Zhang and Wu (2015), and dominating sets Nacher and Akutsu (2012); Gast et al. (2015); Shan et al. (2017). Although there have been concerted efforts to understanding the MIS problem in general, there has been significantly less work focused on the MIS problem for power-law graphs Ferrante et al. (2008). In particular, exact result about the independence number and the number of all MISs in a power-law graph is still lacking, despite the fact that exact result is helpful for testing heuristic algorithms. Moreover, the influence of scale-free behavior on the MIS problem is not well understood, although it is suggested to play an important role in the MIS problem.

The ubiquity of power-law phenomenon makes it interesting to uncover the dependence of MISs on the scale-free feature, which is helpful for understanding the applications of MIS problem. In this paper, we study the independence number and the number of maximum independent sets in a scale-free graph, called pseudofractal scale-free web Dorogovtsev et al. (2002); Zhang et al. (2009), and the Sierpiński gasket. Both networks are deterministic and have the same number of vertices and edges. Note that since determining the independence number and counting all maximum independent sets in a general graph are formidable, we choose these two exactly tractable graphs. This is a fundamental route of research for NP-hard and #P-complete problems. For example, Lovász Lovász and Plummer (1986) pointed out that it is of great interest to find specific graphs for which the matching problem can be exactly solved, since the problem in general graphs is NP-hard.

By using an analytic technique based on a decimation procedure Kneževic and Vannimenus (1986), we find the exact independence number and the number of all possible maximum independent sets for both studied graphs. The independence number of the pseudofractal scale-free web is as twice as the one associated with the Sierpiński gasket. In addition to this difference, there is a unique maximum independent set in the pseudofractal scale-free web, while the number of all maximum independent sets in the Sierpiński gasket increases as an exponential function of the number of vertices.

2 Independence number and the number of maximum independent sets in pseudofractal scale-free web

In this section, we study the independence number in the pseudofractal scale-free web, and demonstrate that its maximum independent set is unique.

2.1 Network construction and properties

The pseudofractal scale-free web Dorogovtsev et al. (2002); Zhang et al. (2009) is constructed in an iterative way. Let , , denote the network after iterations. When , is a triangle. For , is obtained by adding, for every edge in a new vertex connected to and . Figure 1 illustrates the networks for the first several iterations. By construction, the total number of edges in is .

Figure 1: The first three iterations of the scale-free graph.

The network displays the striking properties observed in most real-life networks. First, it is scale-free, since the degree of its vertices obeys a power law distribution  Dorogovtsev et al. (2002)

, implying that the probability of a vertex chosen randomly having degree

is approximately . Moreover, it is small-world, with its average distance growing logarithmically with the number of vertices Dorogovtsev et al. (2002); Zhang et al. (2007). Finally, it is highly clustered, with its average clustering coefficient converging to .

Of particular interest is the self-similarity of network , which is another ubiquitous property of real networks Song et al. (2005). For , the three vertices generated at have the highest degree, which are called hub vertices, and are denoted by , , and , respectively. The self-similar feature of the network can be seen from another construction approach Zhang et al. (2007). Given the network , can be obtained by joining three copies of at their hub vertices, see Fig. 2. Let , , be three replicas of , and denote the three hub vertices of by , , and , respectively. Then, can be obtained by merging , , with (resp. , ) and (resp. , ) being identified as the hub vertex (resp. , ) in .

Figure 2: Alternative construction of the scale-free network.

Let stand for the total number of vertices in . By the second construction of the network, satisfies relation , which together with the initial value , is solved to give .

2.2 Independence number and the number of maximum independent sets

Let denote the independence number of network . To determine , we introduce some intermediate quantities. Since the three hub vertices in are connected to each other, any independent set of

contains at most one hub vertex. We classify all independent sets of

into two subsets and . represents those independent sets with no hub vertex, while denotes the remaining independent sets, with each having exactly one hub vertex. Let , , be the subset of , where each independent set has the largest cardinality (number of vertices), denoted by . By definition, the independence number of network , , is .

The two quantities and can be evaluated by using the self-similar structure of the network.

Lemma 2.1

For two successive generation networks and , ,

(1)
(2)

Proof.   By definition, , , is the cardinality of an independent set in . Below, we will show that both and can be constructed iteratively from and . Then, and can be obtained from and . We now establish the recursive relations for and .

We first prove graphically Eq. (1) .

Notice that consists of three copies of , , . By definition, for any independent set in , the three hub vertices of do not belong to , implying that the corresponding six identified hub vertices of , , are not in , see Fig. 2. Therefore, we can construct set from and by considering whether the hub vertices of , , are in or not. Fig. 3 illustrates all possible configurations of independent sets that include as subsets. From Fig. 3, we obtain

Similarly we can prove Eq. (2), the graphical representation of which is shown in Fig. 4.  

Figure 3: Illustration of all possible configurations of independent sets of , which contain . Only the hub vertices of , , are shown. Filled vertices are in the independent sets, while open vertices are not.
Figure 4: Illustration of all possible configurations of independent sets of , which contain .
Lemma 2.2

For network , , .

Proof.   We prove this lemma by mathematical induction on . For , we obtain , by hand. Thus, the basis step holds immediately.

Assume that the statement holds for (). Then, according to Eq. (1), . By induction hypothesis, we have

(3)

In an analogous way, we obtain relation

(4)

By comparing Eqs. (3) and (4) and using the induction hypothesis , we have . Thus, the lemma is true for .  

Theorem 2.3

The independence number of network , , is

(5)

Proof.   Lemma 2.2 indicates that any maximum independent vertex set of contains no hub vertices. By Eq. (3), we obtain

(6)

Considering the initial condition , the above equation is solved to give the result.  

Corollary 2.4

The largest number of vertices in an independent vertex set of , , which contains exactly hub vertex, is

(7)

Proof.   By Eqs. (5) and (6), we derive . Using Eq. (4), we obtain the following recursive equation for :

(8)

which together with the boundary condition is solved to yield Eq. (7).  

Theorem 2.5

For network , , there is a unique maximum independent set.

Proof.   Eq. (6) and Fig. 3 mean that for any maximum independent set of is actually the union of maximum independent sets, , of the three copies of (i.e. , , and ) constituting . Thus, any maximum independent set of is determined by those of , , and . Because the maximum independent set of is unique, there is a unique maximum independent set for for all . Furthermore, the unique maximum independent set of , , is in fact the set of all vertices that are generated at the -th iteration.  

Theorem 2.5 indicates that the pseudofractal scale-free web is a unique independence graph.

3 Independence number and the number of maximum independent sets in Sierpiński gasket

In this section, we consider the independence number and the number of maximum independent sets in the Sierpiński gasket, and compare the results with those of the pseudofractal scale-free web, with an aim to unveil the effect of network structure, in particular the scale-free property, on the independence number and the number of maximum independent sets.

3.1 Construction of Sierpiński gasket

The Sierpiński gasket is also constructed iteratively. Let , , represent the -generation graph. For , is an equilateral triangle with three vertices and three edges. For , perform a bisection of the three edges of forming four smaller replicas of the original equilateral triangle, and remove the central downward pointing equilateral triangle to get . For , is obtained from by performing the above two operations for each triangle in . Fig. 5 illustrates the first several iterations of the Sierpiński gaskets for .

Figure 5: The first three generations of the Sierpiński gasket.

Both the number of vertices and the number of edges in the Sierpiński gasket are the same as those for the scale-free network , which are equal to and , respectively.

In contrast to the inhomogeneity of , the Sierpiński gasket is homogeneous. The degree of vertices in is equal to 4, except the topmost vertex , the leftmost vertex , and the rightmost vertex , the degree of which is 2. These three vertices with degree 2 are called outmost vertices hereafter.

Figure 6: Alternative construction of the Sierpiński gasket.

Analogously to the scale-free network , the Sierpiński gasket also exhibits the self-similar property, which suggests an alternative construction way of the graph. Given the th generation graph , the th generation graph can be obtained by amalgamating three copies of at their outmost vertices, see Fig. 6. Let , , represent three copies of . And denote the three outmost vertices of by , , and , respectively. Then, can be obtained by coalescing , , with , , and being the outmost vertices , , and of .

3.2 Independence number

In this case without causing confusion, we employ the same notation as those for to study related quantities for the Sierpiński gasket . Let be the independence number of . Note that all independent sets of can be sorted into four types: , , , and , where , , stands for those independent sets, each of which includes exactly outmost vertices of . Let , , denote the subsets of , each independent set in which has the largest cardinality, denoted by . Then, the independence number of the Sierpiński gasket , , is . Therefore, to determine for , one can alternatively determine , , which can be solved by establishing some relations between them, based on the self-similar architecture of the Sierpiński gasket.

Lemma 3.1

For any integer , the following relations hold.

(9)
(10)
(11)
(12)

Proof.   This lemma can be proved graphically. Figs. 7-10 illustrate the graphical representations from Eq. (9) to Eq. (12).  

Figure 7: Illustration of all possible configurations of independent sets of , which contain . Only the outmost vertices of , , are shown. Filled vertices are in the independent sets, while open vertices are not.
Figure 8: Illustration of all possible configurations of independent sets of , which contain . Note that here we only illustrate the independent sets, each of which only includes but excludes , and . Since , , and are equivalent to each other, we omit other independent sets, including (resp. ) but excluding and (resp. and ) .
Figure 9: Illustration of all possible configurations of independent sets of , which contain . Note that here we only illustrate the independent sets, each of which includes two outmost vertices and , but excludes the outmost vertex . Similarly, we can illustrate those independent sets, each including and (resp. and ), but excluding (resp. ) .
Figure 10: Illustration of all possible configurations of independent sets of , which contain .
Lemma 3.2

For arbitrary , .

Proof.   We prove this lemma by induction.

For , it is easy to check that , , , and . Thus, the result holds for .

Let us suppose that the statement is true for , . For , by induction assumption and Lemma 3.1, it is not difficult to check that the relation is true.  

Theorem 3.3

The independence number of the Sierpiński gasket , , is .

Proof.   According to Lemmas 3.1 and 3.2, we obtain . Considering , it is obvious that holds for all .  

Theorems 2.3 and Theorem 3.3 show that the independence number of the Sierpiński gasket is larger than the one corresponding to the pseudofracal scale-free web , with the former being as half as the latter for large .

Corollary 3.4

The largest possible number of vertices in an independent vertex set of , , which contains exactly , and outmost vertices, is , , and , respectively.

Proof.   Theorem 3.3 shows . From Lemma 3.2, we obtain and . Then, the results are obtained immediately.  

3.3 The number of maximum independent sets

In comparison with the scale-free network with a unique maximum independent set, the number of maximum independent sets of increases exponentially with the number of vertices.

Theorem 3.5

For , the number of maximum independent sets of the Sierpiński gasket is .

Proof.   Let denote the number of maximum independent sets of the Sierpiński gasket . Let be the number of independent sets of with maximum number of vertices, including only but excluding , and . For the initial condition , we have , . For , we can prove that the two quantities and obey the following relations:

(13)
(14)

We first prove Eq. (13). By definition, is the number of different maximum independent sets for , each of which contains all the three outmost vertices of . According to Lemma 3.2 and Fig. 10, the two configurations in Fig. 10 which maximize (and thus ) are when , , contains exactly one outmost vertex and when it contains the three outmost vertices. Then, we can establish Eq. (13) by using the rotational symmetry of the Sierpiński gasket.

Eq. (14) can be proved analogously by using Lemma 3.2 and Fig. 8.

Since and , Eqs. (13) and (14) show that for all . Then, we obtain a recursion relation for as , which together with the initial value is solved to yield .  

Acknowledgements

This work is supported by the National Natural Science Foundation of China under Grant No. 11275049.

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