 # Maximum matchings and minimum dominating sets in Apollonian networks and extended Tower of Hanoi graphs

The Apollonian networks display the remarkable power-law and small-world properties as observed in most realistic networked systems. Their dual graphs are extended Tower of Hanoi graphs, which are obtained from the Tower of Hanoi graphs by adding a special vertex linked to all its three extreme vertices. In this paper, we study analytically maximum matchings and minimum dominating sets in Apollonian networks and their dual graph- s, both of which have found vast applications in various fields, e.g. structural controllability of complex networks. For both networks, we determine their matching number, domination number, the number of maximum matchings, as well as the number of minimum dominating sets.

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## 1 Introduction

Let be an unweighted connected graph with vertex set of vertices and edge set of edges. A matching of graph is a subset of , in which no two edges are incident to a common vertex. A vertex linked to an edge in is said to be covered by , otherwise it is vacant. A maximum matching is a matching of maximum cardinality, with a perfect matching being a particular case covering all the vertices. The cardinality of a maximum matching is called matching number of graph . A dominating set of a graph is a subset of , such that every vertex in is connected to at least one vertex in . For a set with the smallest cardinality, we call it a minimum dominating set (MDS), the cardinality of which is called domination number of graph .

Both the maximum matching problem Lovász and Plummer (1986) and MDS problem Haynes et al. (1998) have found numerous practical applications in different areas. For example, matching number and the number of maximum matchings are relevant in lattice statistics Montroll (1964), chemistry Vukičević (2011), while MDS problem is closely related to routing on ad hoc wireless networks Wu (2002)

and multi-document summarization in sentence graphs

Shen and Li (2010). Very recently, it was shown that maximum matchings and MDS are important analysis tools for structural controllability of complex networks Liu and Barabási (2016), based on vertex Liu et al. (2011) or edge Nepusz and Vicsek (2012) dynamics. In the context of vertex dynamics, the minimum number of driving vertices necessary to control the whole network and the possible configurations for sets of driving vertices are, respectively, determined by matching number and the number of maximum matchings in a bipartite graph derived from the original network Liu et al. (2011). In the aspect of edge dynamics, the structural controllability problem can be reduced to determining domination number and the number of MDSs in the original network Nacher and Akutsu (2012, 2013), which has an advantage of needing relatively small control energy Klickstein et al. (2017).

Due to the wide applications, it is of theoretical and practical significance to study matching number and domination number in networks, as well as the number of maximum matchings and MDSs. In the past years, concerted efforts have been devoted to developing algorithms for the problems related to maximum matchings Kuo (2004); Yan et al. (2005); Yan and Zhang (2005, 2008); Teufl and Wagner (2009); Chebolu et al. (2010); Yuster (2013); Zhang and Wu (2015); Meghanathan (2016); Li and Zhang (2017) and minimum dominating sets Fomin et al. (2008); Hedar and Ismail (2012); da Fonseca et al. (2014); Gast et al. (2015); Couturier et al. (2015); Shan et al. (2017) from the community of mathematics and theoretical computer science. However, solving these problems is a challenge and often computationally intractable. For example, finding a MDS of a graph is NP-hard Haynes et al. (1998); enumerating maximum matchings in a generic graph is arduous Valiant (1979a, b), which is #P-complete even in a bipartite graph Valiant (1979b). At present, the maximum matching and MDS problems continue to be an active research object Assadi et al. (2017); Golovach et al. (2016).

Since determining matching number and domination number, especially counting all maximum matchings and minimum dominating sets in a general graph are formidable, it is therefore of much interest to construct or find particular graphs for which the maximum matching and MDS problem can be solved exactly Lovász and Plummer (1986). In this paper, we study maximum matchings and minimum dominating sets in Apollonian networks Andrade Jr et al. (2005) and their dual graphs—extended Tower of Hanoi graphs, which are obtained from the Tower of Hanoi graphs by adding a special vertex connected to all its three extreme vertices Klavžar and Mohar (2005). The Apollonian networks are very useful models capturing simultaneously the remarkable scale-free Barabási and Albert (1999) small-world Watts and Strogatz (1998) properties in realistic networks Newman (2003), and have found many applications Andrade Jr et al. (2005). The deterministic construction of Apollonian networks and their dual graphs allows to treat analytically their topological and combinatorial properties. For both networks, by using the decimation technique for self-similar networks Chang et al. (2007), we find their matching number and domination number, and determine the number of different maximum matchings and minimum dominating sets.

## 2 Constructions and properties of Apollonian networks and extended Tower of Hanoi graphs

In this section, we give a brief introduction to constructions and properties of Apollonian networks and extended Tower of Hanoi graphs—the dual of Apollonian networks.

### 2.1 Construction means and structural properties of Apollonian networks

The Apollonian networks are generated by an iterative approach Andrade Jr et al. (2005).

###### Definition

Let , , denote the Apollonian network after iterations, with and being the vertex set and the edge set, respectively. Then, is constructed as follows:

For , is an equilateral triangle.

For , is obtained from by performing the following operation: For each existing triangle in that was generated at the th iteration, a new vertex is created and connected to the three vertices of this triangle.

Fig. 1 illustrates the construction process of Apollonian networks for the first several iterations.

The Apollonian networks are self-similar, which suggests an alternative construction approach Zhang et al. (2008). For network , we call the three initial vertices in the outmost vertices denoted by , , and , respectively, while call the vertex generated at the first iteration the center vertex, denoted by . Then, the Apollonian networks can be generated in another way as shown in Fig. 2.

###### Definition

Given , , is obtained from by performing the following merging operations:

(i) Joining three copies of , denoted by , , the three outmost vertices of which are denoted by , , and (), with , , and in corresponding to , , and in .

(ii) Identifying and , and , and and as the outmost vertex , , and of ; and identifying , and as the center vertex of .

Let and denote, respectively, the number of vertices and edges in . By construction, and obey relations and . With the initial conditions and , we have and . Thus,

is odd for odd

, and is even for even .

The Apollonian networks display the striking power-law and small-world characteristics of real networks Andrade Jr et al. (2005). Their vertex degree obeys a power-law distribution . Their mean shortest distance of all pairs of vertices increases logarithmically with the number of vertices Zhang et al. (2008). Thus, the Apollonian networks are good models mimicking real networks, and have received considerable attention from the scientific community Zhang et al. (2013, 2014); Zhang and Mahmoud (2016).

### 2.2 Constructions and properties of Tower of Hanoi graphs and their extension

In addition to the remarkable scale-free small-world properties, other intrinsic interest of the Apollonian networks is the relevance to the Tower of Hanoi graphs, that is, the dual graphs of Apollonian networks are extended Tower of Hanoi graphs Zhang et al. (2013).

Before introducing the dual of Apollonian networks, we first introduce the construction of the Tower of Hanoi graphs, which are derived from the Tower of Hanoi puzzle with discs Hinz et al. (2013). The puzzle consists of disks of different sizes and three pegs (towers), labeled by 0, 1, and 2, respectively. Initially, all

disks are stacked on peg 0 ordered by size, with the largest on the bottom and the smallest on the top. The task is to move all disks to peg 2 complying with the following Hanoi rules: (i) At each time step only one topmost disk may be moved; (ii) At any moment, no disk can stack on a smaller one.

A convenient and direct representation of the Tower of Hanoi problem is the graph representation. The set of configurations of the puzzle corresponds to the set of vertices. And the edges specify the legal moves of the game that obey the above rules. The resultant graphs are called the Tower of Hanoi graphs, commonly abbreviated to the Hanoi graphs. Each vertex in the graphs can be labeled by an -tuple or in the regular expression form , where represents the peg on which the th smallest disk resides. Then the labels of any pair of adjacent vertices have the Gray property, that is, they differ in exactly one position Savage (1997). We use to denote the Hanoi graph of discs, where is the vertex set, and is the edge set. Figure 3 shows , and .

In graph , there are vertices. The three vertices , , and , are called extreme vertices, each of which corresponds to a state/configuration with all disks on one of the pegs. Henceforth we use , , and to represent, respectively, the three extreme vertices , , and in . Every extreme vertex has neighbors, while the degree of the remaining vertices is 3. Thus, the number of edges in is .

Note that , , can be obtained from three copies of joined by three edges, each one connecting a pair of extreme vertices from two different replicas of denoted by , and , see Fig. 4. The properties of the Hanoi graphs have been extensively studied Hinz et al. (2013, 2017).

The extended Tower of Hanoi graph , is obtained from by adding a special vertex and three edges linking to the three extreme vertices of . In graph , the degree of every vertex is 3. Thus, the number of vertices and edges in is and , respectively. For odd , is a multiple of 4. Fig. 5 shows an extended Tower of Hanoi graph . It has been shown that extended Tower of Hanoi graph , , is isomorphic to the dual of Apollonian network  Zhang et al. (2013). Figure 5: Illustration of an extended Tower of Hanoi graph S+3.

## 3 Maximum matchings in Apollonian networks and extended Tower of Hanoi graphs

In this section, we study the matching number and the number of maximum matchings in Apollonian networks and extended Tower of Hanoi graphs. We will show that there exist no perfect matchings in Apollonian networks when , but there always exist perfect matchings in extended Tower of Hanoi graphs for all . Moreover, we will determine the number of distinct maximum matchings in both networks.

### 3.1 Matching number and the number of maximum matchings in Apollonian networks

We first study the matching number and the number of maximum matchings in Apollonian networks.

#### 3.1.1 Matching number

Although for a general graph, its matching number is not easy to determine, for Apollonian network , we can obtain it by using its self-similar structure. Let be the matching number for . In order to find , we introduce some useful quantities. According to the number of covered outmost vertices, all matchings of can be sorted into four classes: , , with representing the set of matchings covering exactly outmost vertices. Let , be subset of consisting of all elements with maximum size (cardinality). Let , , denote the size of each matching in .

###### Lemma

The matching number of network , , satisfies

.

After reducing the problem of determining to computing and , we next evaluate their values.

###### Lemma

For two successive Apollonian networks and , , the following relations hold.

 β0n+1=max{3β0n,2β0n+β1n}, (1)
 β1n+1=max{2β0n+β1n,2β0n+β2n,β0n+2β1n}, (2)
 β2n+1=max{2β0n+β3n,β0n+β1n+β2n,3β1n,β0n+2β1n,2β0n+β2n}, (3)
 β3n+1=max{β0n+β1n+β2n,β0n+β1n+β3n,β0n+2β2n,3β1n,2β1n+β2n}. (4)

###### Proof

This lemma can be proved graphically. Note that each set ,

, can be further classified into two groups of matchings: one group includes those matchings covering the center vertex in

, while the other group contains those matchings with the center vertex vacant. According to the classification, Figs. 678 and 9 show, respectively, configurations of matchings for belonging to , , that comprise all the matchings in . In Figs. 678 and 9, only the outmost and center vertices are shown explicitly, with filled circles being covered vertices and empty circles being vacant vertices. From Figs. 678 and 9, we can establish the relations in Eqs. (1), (2), (3) and (4). Figure 6: Configurations of matchings for An+1 belonging to Ω0n+1, which contain all matchings in Φ0n+1. Figure 7: Configurations of matchings for An+1 belonging to Ω1n+1, which contain all matchings in Φ1n+1. In view of the symmetry, we only illustrate those matchings of An+1, for each of which X is covered, while Y and Z are vacant, but omit matchings that cover only Y (or Z), but not cover X and Z (or X and Y). Figure 8: Configurations of matchings for An+1 belonging to Ω2n+1, which contain all matchings in Φ2n+1. By symmetry, as in Fig. 7, we only illustrate those matchings of An+1, for each of which X and Y are covered, while Z is vacant. Figure 9: Configurations of matchings for An+1 belonging to Ω3n+1, which contain all matchings in Φ3n+1.

For small , , , and can by directly computed by hand. For example, for , we have , , and , respectively. For , the following lemma establishes the relations between , , and .

###### Lemma

For Apollonian network , , quantities , satisfy the following relations:

 β0n+3=β1n+2=β2n+1=β3n. (5)

###### Proof

By induction. When , the result is true.

Let us suppose that the relations hold true for , i.e., . For , by Lemma 3.1.1 we get . This completes the proof.

###### Theorem

For Apollonian network , , its matching number is

 βn=3n−1+52. (6)

###### Proof

Lemma 3.1.1 combined with Lemma 3.1.1 shows that . From Lemma 3.1.1, we obtain the recursive relation governing and as , which, together with the initial value , is solved to yield .

The matching number for Apollonian network , , is much smaller than half the number of vertices , indicating that no perfect matching exists in .

###### Corollary

For Apollonian network , , the maximum size of a matching covering exactly , , and outmost vertices, is

 β0n=3n−1−12, (7)
 β1n=3n−1+12 (8)

and

 β2n=3n−1+32, (9)

respectively.

###### Proof

By Eqs. (5) and (6), the result is concluded immediately.

#### 3.1.2 Number of maximum matchings

Let be the number of maximum matchings of . To determine , we introduce three more quantities. Let be the number of matchings of that are maximum among those matchings for which all the three outmost vertices , , and are vacant. Let be the number of matchings of that are maximum among those matchings for which is covered, while and are vacant. By symmetry, equals the number of matchings of that are maximum among those matchings for which () is covered, while and ( and ) are vacant. Let be the number of matchings of that are maximum among those matchings for which is vacant, while and are covered. Obviously, is equal to the number of matchings of that are maximum among those matchings for which () is vacant, while and ( and ) are covered. For small , quantities , , , and can be easily determined by using a computer. For example, in the case of , , , , and . For large , they can be determined recursively as follows.

###### Theorem

For Apollonian network , , the four quantities , , , and can be recursively determined according to the following relations:

 φn+1=3θnφ2n, (10)
 θn+1=2ϕnφ2n+4θ2nφn, (11)
 ϕn+1=τnφ2n+8ϕnθnφn+3θ3n, (12)
 τn+1=6τnθnφn+6φnϕ2n+12θ2nϕn, (13)

with initial values , , , and .

###### Proof

First we show that all matchings listed in Figs. 6,  7,  8, and  9 are truly disjoint for . For this purpose, it is sufficient to show that when , for any matching of belonging to , , no edge connecting the outmost vertex and another outmost vertex (or ) in () is in . This can be proved as follows.

We only prove the case that such an edge is inexistent in any matching in . Otherwise, suppose that the edge linked to and is in . We distinguish two cases: (i) is vacant; (ii) is covered by an edge in . For case (i), all edges belonging to and at the same time constitute a matching of that is in . If we delete , then all the remaining edges simultaneously belonging to and form a matching of that is in . This means , in contradiction with the fact obtained in Lemma 3.1.1. Analogously, we can prove case (ii).

In a similar argument, we can prove that such an edge does not exist in any matching in or .

Then, to prove Eqs. (10), (11), (12), and (13), it suffices to count all cases that yield maximum matchings in , , listed in Figs. 678 and 9, respectively.

We here only prove Eq. (12), since the other three equations can be proved in a similar way.

According to Eq. (3) and Lemma 3.1.1, all the possible configurations of maximum matchings with size are those having size , or . From Fig. 8, we can see that among all maximum matchings of covering and but not , the number of those with size , , is equal to , , and , respectively. By definition of , Eq. (12) holds.

Theorem 3.1.2 shows that the number of maximum matchings of Apollonian network can be calculated in time.

Applying Eqs. (10)-(13), the values of , , , and can be determined for small as listed in Table 1. However, it is very difficult to obtain closed-form expressions for these quantities.

#### 3.1.3 Asymptotic growth constant for the number of maximum matchings

Table 1 shows that number of maximum matchings grows exponentially with . Let , then we can define a constant describing this exponential growth:

 Z(A)=limn→∞Z(An)=limn→∞lnτnVn. (14)

Below we will show that this limit exists. To evaluate the asymptotic growth constant in Eq. (14), we need the following lemma.

###### Lemma

For , the quantities , , , and , describing the number of different maximum matchings in Apollonian network under certain conditions, obey the following relation:

 φn≤θn≤ϕn≤τn. (15)

###### Proof

We prove this lemma by induction on . Table 1 shows that for , the result holds. Let us assume that the lemma is true for , that is, . For , according to induction assumption and Theorem 3.1.2, we obtain . This completes the proof.

In order to estimate the asymptotic growth constant

, we define several ratios: , , and .

###### Lemma

Let and . Then for , the three ratios , , and obey following relations: , , and .

###### Proof

It is easy to see that are all constantly positive. By definition,

 αn+1=φn+1θn+1=3θnφ2n2ϕnφ2n+4θ2nφn≤34⋅φnθn=34αn.

Thus, is a decreasing positive sequence that converges to zero, implying as .

Next we bound and . By definitions, we have

 ηn+1=θn+1ϕn+1=2ϕnφ2n+4θ2nφnτnφ2n+8ϕnθnφn+3θ3n≤2ϕnθnφn+2ϕnθnφn+2θ3n8ϕnθnφn+3θ3n≤23

and

 λn+1=τn+1ϕn+1= 6τnθnφn+6φnϕ2n+12θ2nϕnτnφ2n+8ϕnθnφn+3θ3n≥6τnθnφn+6φnϕ2n+12θ2nϕnτnθnφn+8ϕnθ2n+3θ2nϕn ≥ 6τnθnφn+12θ2nϕnτnθnφn+11θ2nϕn≥1211,

which, together with the initial values and , show that , and .

The above properties of related ratios are given in a loose way. One can provide a tighter bound for , for example. However, they are enough to find the asymptotic growth constant .

###### Theorem

For Apollonian network , has a limit when is sufficiently large, and the asymptotic growth constant for the number of maximum matchings is .

###### Proof

By Eqs. (10), (11),  (12),  (13), we obtain

 φn+1=3αnφ3n (16)

and

 τn+1=6τ3n(ηnλn⋅αnηnλn+αnηnλ3n+2(ηnλn)21λn). (17)

Denote and . According to Lemma 3.1.3, one has . From Eqs. (16) and (17), we obtain that for ,

 φn≥pmφ3n−1≥pmp3mφ9n−2≥⋯≥p3n−m−12mφ3n−mm, (18)
 τn≤⋯≤q3n−m−12τ3n−mm. (19)

Taking logarithm on both sides of Eqs. (18) and (19) gives rise to

 logφn≥3n−mlogφm+3n−m−12logpm, (20)
 logτn≤3n−mlogτm+3n−m−12logq. (21)

Notice that for , as stated in Lemma 3.1.3. By definition, . Then, for ,

 2logφm3m+logpm3m≤Z(An)≤2logτm3m+logq3m. (22)

Because , then as . Therefore, the difference of the leftmost and rightmost sides in Eq. (22) converges to zero as , meaning that has a limit when .

The convergence of the upper and lower bounds for in Eq. (22) is rapid. For example, when is 7, the difference between the upper and lower bounds is less than . Note that when , the difference of the exact values for the upper and lower bounds is approximately equal to , implying that both bounds are a good approximate of the limit. In other words, even for small , the upper and lower bounds in Eq. (22) converge to the quoted value .

### 3.2 Matching number and the number of maximum matchings in extended Tower of Hanoi graphs

We proceed to study the matching number and the number of maximum matchings in extended Tower of Hanoi graphs, the dual of Apollonian networks. We will show that in contrast to Apollonian networks, there always exists a perfect matching in extended Tower of Hanoi graphs. Moreover, we will determine the number of perfect matchings in extended Tower of Hanoi graphs.

#### 3.2.1 Matching number

In the case without inducing confusion, we employ the same notation as those for studied in the preceding subsection. Let stand for the matching number of . Let be a subset of for graph , we use to denote a subgraph of , which is obtained from by deleting those vertices in and edges adjacent to any vertex in . Then, is isomorphic to .

###### Theorem

In the extended Tower of Hanoi graph , , a perfect matching always exists. Thus, the matching number of is

 βn=Vn2=3n+12. (23)

###### Proof

It suffices to show that the graph , , has a perfect matching. Then the second result immediately follows from the fact that .

We first prove the existence of a perfect matching in , , by induction on . When , it is easy to check that a perfect matching exists in . Thus the result holds for the base case. Assume that for , there exists a perfect matching in , which means that there exists at least a perfect matching in the three subgraphs , , and of . Then there exists a perfect matching in , , and , see Fig. 10, where , , and denote the extreme vertices of , , forming . By adding to the edge connected the special vertex and the vacant extreme vertex , , or , leads to a perfect matching of , as shown in Fig. 10. Figure 10: Illustration of existence of a perfect matching in S+k+1 for k≥2. The cross vertex denotes the removed extreme vertex ck+1 from Hk+1. A perfect matching for Hk+1∖{ck+1} can be obtained by adding one thick edges linked to the two extreme vertices c1k in H1k and c2k and H2k to the union of perfect matchings for H1k∖{c1k}, H2k∖{c2k}, and H3k∖{c3k}. Adding to a perfect matching of Hk+1∖{ck+1} one thick edge connecting to the extreme vertex ck+1 and the special vertex s gives a perfect matching for S+k+1.

#### 3.2.2 Number of perfect matchings

Let , , be the maximum size of matchings of the subgraph of obtained from by deleting exactly extreme vertices and the edges attaching to these extreme vertices. For any matching of with at least one vacant extreme vertex, we can obtain a matching of from by adding to it an edge adjacent to a vacant extreme vertex and the special vertex .

Note that for any perfect matching in , the number of matching edges in its subgraph , , must be or . Otherwise, we suppose that in some , the number of matching edges is or , then there must be one vacant non-extreme vertex in , meaning that cannot be a perfect matching of .

Next, we show that in graph , any matching with size is indeed a perfect matching.

###### Lemma

For , there exists a perfect matching in the subgraph of the extended Tower of Hanoi graph , and its matching number is

 β0n=3n−32. (24)

###### Proof

We prove this lemma by induction on . When , the three edges connecting the three pairs of extreme vertices in , , form a perfect matching of . Thus, the base case holds. Suppose that for , there exists a perfect matching in . Then for , we can obtain a perfect matching of by adding to the perfect matchings of , , three edges connecting three pairs of extreme vertices in , See Fig. 11. Then, the number of edges in a perfect matching of is . Figure 11: Illustration of existence of a perfect matching in Hk+1∖{ak+1,bk+1,ck+1} for k≥2. The cross vertices denote those removed extreme vertices from Hk+1.

Let be the n