Spanning Trees and Domination in Hypercubes

TL;DR

This paper investigates the properties of spanning trees and dominating sets in hypercube graphs, revealing asymptotic behaviors and constructing specific trees using coding theory and expansion techniques.

Contribution

It provides new asymptotic estimates for the maximum leaves in spanning trees and the minimum connected dominating sets in hypercubes, employing Hamming codes and expansion methods.

Findings

Maximum leaves in spanning trees of hypercubes grow exponentially with n.

Minimum connected dominating set size is approximately 2^n/n.

Constructive methods for leafy spanning trees using coding theory.

Abstract

Let $L(G)$ denote the maximum number of leaves in any spanning tree of a connected graph $G$. We show the (known) result that for the $n$-cube $Q_n$, $L(Q_n) \sim 2^n = |V(Q_n)|$ as $n\rightarrow \infty$. Examining this more carefully, consider the minimum size of a connected dominating set of vertices $\gamma_c(Q_n)$, which is $2^n-L(Q_n)$ for $n\ge2$. We show that $\gamma_c(Q_n)\sim 2^n/n$. We use Hamming codes and an "expansion" method to construct leafy spanning trees in $Q_n$.

Equations

$$L(G)+{\gamma }_c(G)=|V(G)|.$$

$$1\le \gamma (G)\le {\gamma }_c(G)\le |V|.$$

$$L(Q_n)=(1-\frac{1}{n}+o(\frac{1}{n}))2^n.$$

$${\gamma }_c(G)\le 3\gamma (G)-2.$$

$$\gamma (Q_n)\le 2^j\frac{2^N}{N+1}=\frac{N+j}{N+1}\frac{2^n}{n}<2\frac{2^n}{n}.$$

$$\gamma (Q_n)/2^n<2/n\to 0,$$

$${\gamma }_c(Q_n)<3\gamma (Q_n)<6\frac{2^n}{n}.$$

$$\frac{{\gamma }_c(Q_n)}{2^n}<\frac{6}{n}\to 0,$$

$${\gamma }_c(Q_n)$$

$$<(2^j+2)|C|$$

$$=(2^j+2)(2^{N-k})$$

$$=(2^n+2^{N+1})/2^k.$$

$$\frac{{\gamma }_c(Q_n)}{2^n/n}<\left(1+\frac{1}{2^{j-1}}\right)\left(1+\frac{j-1}{2^k}\right).$$

$${\gamma }_c(Q_n)<(2^j+2)\gamma (Q_N).$$

$$\underset{n\to \infty }{\lim }\frac{\gamma (Q_n)}{2^n/n}=1.$$

$$\underset{n\to \infty }{\lim }\frac{{\gamma }_c(Q_n)}{2^n/n}=1.$$

$$\frac{\gamma (Q_N)}{2^N/N}<1+\epsilon .$$

$$\frac{{\gamma }_c(Q_n)}{2^n/n}<(1+\epsilon {)}^2,$$

Full text

1 Introduction

The $n$-cube graph $Q_{n}$ has $2^{n}$ vertices, the strings $a_{1}\ldots a_{n}$ on $n$ bits, where two vertices are adjacent if and only if their strings differ in exactly one coordinate (where one vertex has 0 and the other has 1). The $n$-cube is frequently used as a structure for computer networks, where there are $2^{n}$ processors corresponding to the vertices of $Q_{n}$. An efficient way to connect all of the processors, so that they all communicate with each other, is to take a spanning tree in $Q_{n}$.

With this in mind, S. Bezrukov imagined it would be interesting to construct such spanning trees with many leaves (degree one vertices). At the IWOCA conference (Duluth, 2014), Bezrukov proposed the following problem: Letting $L(G)$ denote the maximum number of leaves in any spanning tree of a connected simple graph $G$, what can one say about $L(Q_{n})$? He shared this problem in notes [2].

For a spanning tree, the non-leaf vertices are connected, so form a tree themselves, which we may think of as the backbone of the tree: All vertices are either in this backbone, or are leaves adjacent to it. Bezrukov’s question then is equivalent to constructing a spanning tree of the hypercube with the smallest backbone.

Notice that the opposite question, finding the minimum number of leaves in a spanning tree, is easy: By a simple induction $Q_{n}$ has a Hamilton path for all $n\geq 1$. This path is a spanning tree with just two leaves. We are interested in the other extreme, maximizing the number of leaves.

Our problem is closely related to the subject of domination in graphs. A subset $W$ of the vertex set $V$ of a graph $G=(V,E)$ is a dominating set if every vertex is either in $W$ or adjacent to some vertex in $W$. The domination number $\gamma(G)$ is the minimum size of any dominating set.

Note that if one pulls off the leaves from a spanning tree $T$ for a connected graph $G=(V,E)$ with at least three vertices, then the remaining vertices $W$ form a dominating set, and, moreover, what remains of $T$ still connects them. That is, $W$ forms a connected dominating set. Conversely, from any connected dominating set we can span them with a tree and attach any other vertices as leaves to obtain a spanning tree. The minimum size of a connected dominating set of $G$ is called the connected domination number $\gamma_{c}(G)$.

We see that maximizing the number of leaves of any spanning tree of such $G$ corresponds to minimizing the size of a connected dominating set. From this discussion we obtain for such $G$

[TABLE]

The simple ordering relationship between these parameters is

[TABLE]

For example, one can readily check that for the four-cycle $Q_{2}$, $\gamma=\gamma_{c}=L=2$, while for the ordinary cube $Q_{3}$, $\gamma=2,\gamma_{c}=L=4$. For larger $n$ more than half the vertices can be leaves.

The earliest paper we can find that investigates the connected domination number of a graph is by Sampathkumar and Walikar (1979)  [15]. Several studies investigate bounding $L(G)$ for classes of graphs $G$, such as those with given minimum degree [16, 8, 12, 9]. Caro et al. [3]study both parameters, and provide more references. Many papers concern algorithms for finding leafy trees (or small connected dominating sets).

Searching online we discovered several papers concerning domination in hypercubes. These were often done independently of other studies. The 1990 dissertation of Jha [10] gives a good general upper bound on $\gamma(Q_{n})$, which is just twice the easy lower bound. Arumugam and Kala [1] (1998) focus on domination in hypercubes. Duckworth et al. [6](2001) give good general bounds on $L(Q_{n})$. It follows that $L(Q_{n})\sim 2^{n}$. It means asymptotically there is a spanning tree for the hypercube in which almost all vertices are leaves. It is nicer to restate their results in terms of connected domination:

Theorem 1.1

[6]**

• •

Lower bound: For $n\geq 2$, $\frac{\gamma_{c}(Q_{n})}{2^{n}}\geq\frac{1}{n}$

• •

Upper bound: As $n\rightarrow\infty$, $\frac{\gamma_{c}(Q_{n})}{2^{n}}\leq(1+o(1))\frac{2}{n}$

Another 2012 study of hypercubes [4] gives values of $\gamma_{c}(Q_{n})$ for small $n$, but unfortunately its formula for general $n$, stated without proof, is far from correct. Mane and Waphare [14] investigate several generalizations of domination numbers of hypercubes. The 2017 Master’s thesis of Kuboň [13] considers domination in hypergraphs, and uses some of the same methods as in this paper.

In the next section, we present simple general lower bounds on $\gamma(Q_{n})$ and $\gamma_{c}(Q_{n})$. In Section 3 we describe the Hamming code construction that gives a “perfect dominating set” for $Q_{n}$ when $n$ is of the form $2^{k}-1$. We give a method to produce a small connected dominating set, given a dominating set, that leads to an upper bound on $\gamma_{c}(Q_{n})$ for $n=2^{k}-1$. A simple inductive method we call doubling is used to give upper bounds on $\gamma(Q_{n})$ and $\gamma_{c}(Q_{n})$ for general $n$ in Section 4.

Where we make new progress is by introducing in Section 5 a new method we call expansion, in which we take a minimum dominating set in each of $2^{j}$ copies of $Q_{N}$ and connect them appropriately to obtain a small connected dominating set in $Q_{n}$, where $n=N+j$. Choosing $N$ and $j$ wisely improves the best previous upper bound on $\gamma_{c}(Q_{n})$ by a factor of 2. Indeed, in Section 6 we settle the leading asymptotic behavior of $\gamma_{c}(Q_{n})$:

Theorem 1.2

As $n\rightarrow\infty$, $\frac{\gamma_{c}(Q_{n})}{2^{n}}=(1+o(1))\frac{1}{n}.$

Restating this in terms of the maximum number of leaves, it means

[TABLE]

We conclude with suggestions for further study and acknowledgements of valuable ideas and support of this project.

2 Domination Lower Bounds

Let us note some easy lower bounds on our domination parameters for the hypercube $Q_{n}$.

Proposition 2.1

• •

For $n\geq 1$, $\gamma(Q_{n})\geq 2^{n}/(n+1).$

• •

For $n\geq 2$, $\gamma_{c}(Q_{n})\geq(2^{n}-2)/(n-1)\geq 2^{n}/n.$

Proof. A single vertex can dominate at most itself and its $n$ neighbors, leading to the lower bound on $\gamma(Q_{n})$.

Next, consider a connected dominating set of $Q_{n}$ of size $c$. There is a tree $T$ on these $c$ vertices using $c-1$ edges from $Q_{n}$. The sum of degrees of these $c$ vertices has $2c-2$ accounted for by $T$. It means that the number of additional vertices (dominated by those in $T$) is at most $nc-2(c-1)$. But there are $2^{n}-c$ vertices besides $T$. Rearranging terms gives the stated inequality on $c$, hence the lower bounds on $\gamma_{c}(Q_{n})$. $\square$

3 Hamming Code

The famous Hamming code gives an elegant construction of a “perfect dominating set” in $Q_{n}$ when $n=2^{k}-1$ for some integer $k\geq 1$. This means it achieves the lower bound on $\gamma(Q_{n})$ in Proposition 2.1. Viewing the vertices of $Q_{n}$ for such an $n$ as $n$-dimensional vectors over $GF(2)$, the code consists of the $2^{n-k}$ vectors in the row space of a $(n-k)\times n$ matrix built as follows: The first $n-k$ columns form the identity matrix, while the rows of the other $k$ columns consist of all $n-k=2^{k}-k-1$ vectors of length $k$ with weight (number of ones) at least 2. The difference between any two vectors in this row space is then a nonzero vector in the row space, and hence a nonempty sum of rows of the matrix. By design, such a sum will always have weight at least three.

Consequently, the $2^{n-k}$ stars in $Q_{n}$ that are centered at the vectors in the row space are disjoint. Each star is a $K_{1,n}$. By counting, we see that these stars partition the vertices of $Q_{n}$. They form a minimum dominating set for $Q_{n}$.

As Bezrukov pointed out when he proposed his problem about $L(Q_{n})$, for such $n$ we only have to add some edges between leaves of different stars to obtain a spanning tree with many leaves. After all, $Q_{n}$ is connected, and all edges not used in the stars are between leaves of stars (different stars, in fact). If we have $c$ components, we need to add $c-1$ edges to obtain a spanning tree; here, $c=2^{n-k}$. At worst, each additional edge costs us two new leaves–it would be less, if we are able to use several edges from the same leaf. When we finish, we have a spanning tree where the non-leaves are the $c$ star centers from the Hamming code, as well as at most $2c-2$ vertices that were star leaves.

In fact, we can use this method for any connected simple graph $G$ to build a spanning tree. Starting from a minimum dominating set of $c$ vertices, the stars centered at those vertices cover the entire vertex set (though in general they are not disjoint, and dominating vertices could even be adjacent). One can add at most $c-1$ edges between stars to create a spanning tree. We obtain this general bound:

Proposition 3.1

Let $G$ be a connected simple graph. Then

[TABLE]

Applying this to our Hamming code construction, we obtain

Proposition 3.2

Let $n=2^{k}-1$, where the integer $k\geq 1$. Then $\gamma(Q_{n})=2^{n-k}=2^{n}/(n+1)$, and $\gamma_{c}(Q_{n})<(3/(n+1))2^{n}.$

For this Hamming code case $n=2^{k}-1$ our tree construction can be viewed this way: Starting from a perfect dominating set in $Q_{n}$, we take the corresponding $C=2^{n}/(n+1)$ stars $K_{1,n}$ and add $C-1$ edges to form a tree with many leaves. Since all edges for the star centers are used already, each edge we add will join leaves from two different stars. At worst, we give up $2(C-1)$ star leaves (they become part of the tree backbone), plus the backbone contains the $C$ star centers. This gives us a connected dominating set of size at most $3C-2\sim 3(2^{n}/n)$.

If we are fortunate, we don’t have to pick two new leaves for each successive additional edge: It could be that one or both leaves are already in the backbone. However, for each of the $C$ stars we must give up at least one leaf, in order that the stars connect in the spanning tree. It means that the connected dominating set we construct will have at least $2C\sim 2(2^{n}/n)$ vertices.

4 Doubling

So far, we have constructed leafy trees in the $n$-cube only when $n$ has the special form $2^{k}-1$. The $(n+1)$-cube can be viewed as built from two copies of $Q_{n}$, with a matching of edges joining the corresponding vertices from each copy. This is true for any value of $n$, not just the special values where the Hamming code exists.

If we take a dominating set for each copy of $Q_{n}$, we get a dominating set for $Q_{n+1}$. Moreover, if we take the same connected dominating set for each copy, it gives a dominating set for $Q_{n+1}$ that is connected. We see this simply by adding the edge joining the two copies of a vertex in the connected dominating set for $Q_{n}$. We record these observations about doubling:

Proposition 4.1

For all $n\geq 1$, $\gamma(Q_{n+1})\leq 2\gamma(Q_{n})$, and $\gamma_{c}(Q_{n+1})\leq 2\gamma_{c}(Q_{n})$.

Now suppose $n$ is between two consecutive values where the Hamming code construction is the last section applies, say $n=N+j$, where $k\geq 1$, $N=2^{k}-1$, and $0\leq j\leq 2^{k}$. We apply the doubling proposition $j$ times, starting from $Q_{N}$, and obtain:

[TABLE]

It follows that

[TABLE]

as $n\rightarrow\infty$. This matches the bound given by Jha [10].

For connected domination we apply Proposition 3.1 and obtain:

[TABLE]

It follows that

[TABLE]

as $n\rightarrow\infty$, confirming our earlier assertion that there are spanning trees for hypercubes with almost all vertices being leaves. Of course, Theorem 1.1 got a better bound than this on $\gamma_{c}(Q_{n})/2^{n}$; Our main result will do even better.

Let us summarize our findings so far. The domination problem for $Q_{n}$ is solved by the Hamming code for $n=N=2^{k}-1$. Then as $n=N+j$ grows with $j,0\leq j\leq 2^{k}$, our upper bound on $\gamma(Q_{n})/(2^{n}/n)$ grows from around 1 to around 2. However, at $j=2^{k}$, we have the next Hamming code case, $n=2^{k+1}-1$, and it is better to switch again to the Hamming code construction. It means we have a sawtooth function upper bound, rising from 1 to 2 as $n$ increases, then abruptly dropping back down to 1 and rising again. Of course, each tooth covers an interval of length about $2^{k}$, so the teeth get wider with $k$.

Owing to our upper bound Proposition 3.1, for connected domination $\gamma_{c}(Q_{n})$ has a similar sawtooth upper bound, but each tooth rises from value 3 to 6.

5 Expansion

We introduce a new method of tree construction that takes advantage of small dominating sets to produce smaller connected dominating sets in $Q_{n}$. This will bring down our upper bound for connected domination, and eventually allow us to solve our problem asymptotically.

For constructing a spanning tree, the Hamming code bound punished us by potentially using up so many leaves to connect the stars. If we repeatedly double the construction, then it repeats this penalty over and over. A better idea could then be to select one copy (or “layer”) of the base hypercube, add edges to connect the stellar clusters in just that layer, and then connect all the copies of each star center to the one in the special layer.

Describing this explicitly, let $N=2^{k}-1$, and $n=N+j$, where $0\leq j\leq 2^{k}$. Partition the vertices of $Q_{n}$ into $2^{j}$ ”layers” according to the last $j$ coordinates of the vertices $(a_{1},\ldots,a_{n})$. Each layer induces a $Q_{N}$, and its vertices are partitioned into $|C|=2^{N-k}$ stars, according to the Hamming code partition of $Q_{N}$. For each star $S$ in the partition of $Q_{N}$, there are $2^{j}$ vertices, one in each layer, that are centers of the stars corresponding to star $S$. The centers all agree in their first $N$ coordinates, so together induce a subgraph $Q_{j}$. By adding $2^{j}-1$ edges these stars (copies of $S$) can be connected into a tree. We now have a forest of $|C|=2^{N-k}$ such trees.

We connect these trees by adding $|C|-1$ edges, which may as well all be in the layer ending with 0’s. Each such edge adds at most two vertices to the connected dominating set we construct. It is similar to how we connected the stars in the Hamming code construction. We record the result of our expansion construction:

Proposition 5.1

Let $n=N+j$, where $N=2^{k}-1$ and $1\leq j\leq 2^{k}$. Then $\gamma_{c}(Q_{n})\leq 2^{j}|C|+2(|C|-1),$ where $C$ is the set of $2^{N-k}$ codewords for the Hamming code in $Q_{N}$.

We have seen that $\gamma_{c}(Q_{n})/2^{n}\geq 1/n$ for all $n\geq 2$. It would be nice if we could find a tree construction for $Q_{n}$ that has so many leaves that its backbone (connected dominating set) comes close to achieving the lower bound, acting asymptotically like a perfect dominating set: What we want is that $\gamma_{c}(Q_{n})/(2^{n}/n)\rightarrow 1$ as $n\rightarrow\infty$. Expansion allows us to come much closer to this goal. Here is what we can show now:

Theorem 5.2

For all $n\geq 1$, $\gamma_{c}(Q_{n})/2^{n}<2/n$. For all $n\geq 3$, $\gamma_{c}(Q_{n})/2^{n}>1/n.$ We have $\liminf_{n\rightarrow\infty}\gamma_{c}(Q_{n})/(2^{n}/n)=1$.

Proof. We have $n,N,K,j$ as above. Proposition 5.1 gives us

[TABLE]

We rewrite this as

[TABLE]

In our range $1\leq j\leq 2^{k}$, the first term in the product on the right starts at 2 and decreases exponentially quickly towards 1. The second term starts at 1 and grows linearly to just below 2 at the end of this range. Throughout this whole range in $j$, the product is at most $2$, giving us the first statement of the theorem.

The second statement, the lower bound on $\gamma_{c}(Q_{n})/2^{n}$, follows easily from Proposition 2.1. For the third statement, we select values of $n$ for which we can show $\gamma_{c}(Q_{n})/(w^{n}/n)$. Specifically, given $k$ take $j=k+1$, so that $n=2^{k}+k$. Then in the upper bound inequality above on $\gamma_{c}(Q_{n})/(2^{n}/n)$, both terms in the product are small (slightly above 1), and their product $\rightarrow 1$ as $k\rightarrow\infty$. The $\liminf$ statement follows. $\square$

An interesting observation is that for $n$ of the form $2^{k}-1$, the Hamming code exists, but the corresponding spanning tree construction for $Q_{n}$ we described earlier only guarantees that $\gamma_{c}(n)/(2^{n}/n)$ is at most $3$ for such $n$. We can do better, constructing a tree that reduces the bound for such $n$ to 2, by taking the Hamming construction for $2^{k-1}-1$ and applying the expansion method with $j=2^{k-1}$. Nevertheless, we are still seeking to do better, aiming to construct trees that bring the bound down to 1 asymptotically.

6 Main result

We have shown how to construct spanning trees for hypercubes $Q_{n}$ with many leaves–the proportion of the $2^{n}$ vertices that are not leaves is at most roughly $2/n$. The idea is to take a Hamming code and then expand.

Now observe that the expansion idea can be used starting from any values of $N$, not just a Hamming code value $2^{k}-1$, and from any dominating set $C$ in $Q_{N}$, to produce a connected dominating set for $Q_{n}$, $n=N+j$: Set $C$ gives a partition of $Q_{N}$ into stars. For each star center (vertex in the dominating set), add edges to connect the $2^{j}$ copies of the vertex. In the original $Q_{N}$ add edges to connect the stars. We now have a spanning tree for $Q_{n}$. Denote by $D$ its backbone, a connected dominating set in $Q_{n}$. We get an upper bound on $|D|$ as in Proposition 5.1. Assuming $|C|$ is minimum-sized, we get that

[TABLE]

Given $n$ large, let $j$ be an integer close to $\log n$ (logarithm base 2), and take $N=n-j$. Then the display above implies that $\gamma_{c}(Q_{n})/(2^{n}/n)$ is bounded above approximately by $\gamma(Q_{n})/(2^{N}/N)$. So an upper bound function for the domination number, shifted to the right by $\log n$, yields an approximate upper bound function on connected domination.

In particular, if it holds that for domination $\gamma(Q_{n})/(2^{n}/n)$ tends towards 1, its lower limit, then the same will be true for the similar expression for connected domination! Fortunately, what we need is proven in the 1997 book Covering Codes by Cohen, Honkala, Litsyn, and Lobstein [5], p.332. They attribute the result to Kabatyanskii and Panchenko [11] (1988). The proof relies on various coding constructions, including $q$-ary Hamming codes for prime powers $q$. It also depends on results on the density of primes.

We include their result on the domination number as the first part of our Main Theorem below. It is restated for convenient comparison to our result for connected domination number, the second part, which can be viewed as a strengthening of the first part.

Theorem 6.1

The domination ratio for hypercubes satisfies [11]

[TABLE]

The connected domination ratio satisfies

[TABLE]

Proof.

As noted above, the first statement is proven in the literature. What is new is the second part, which is a stronger statement. Building on Theorem  it suffices to give an upper bound on $\gamma_{c}(Q_{n})/(2^{n}/n)$ that goes to 1 as $n\rightarrow\infty$. As in the discussion above, given $n$ we take $j$ is close to $\log n$ and $N=n-j$. Given $\varepsilon>0$ we have that for all sufficiently large $n$ (and $N$) that

[TABLE]

Applying this in the discussion above, gives us for all sufficiently large $n$ that

[TABLE]

and the second part follows.

$\square$

Formulating this equivalently in terms of leaves in spanning trees, we obtain:

Corollary 6.2

As $n\rightarrow\infty$, $L(Q_{n})=2^{n}(1-\frac{1}{n}+o(\frac{1}{n})).$

7 Further Study

Here are some ideas for continuing research. We were not able to give a simple enough proof that the domination number that $\gamma(Q_{n})/(2^{n}/n)\rightarrow 1$. We were hoping to give a self-contained proof of our main result. The proof in the literature of this domination result relies on rather technical explicit coding constructions. It would be nice if one could devise an algorithm, or use probabilistic arguments, to prove the existence of dominating sets in the hypercube that are as small as the theorem.

Another question asked by Bezrukov [2] remains open: For $n=2^{k}-1$, starting from the stars given by the Hamming code, how can one add edges to form a tree with the most leaves (the smallest connected dominating set)? We have seen that for large $k$ the connected dominating set will have size between 2 and 3 times $2^{n}/n$. How can one add edges efficiently, to get close to the lower bound?

What can one say about a more general class of graphs? For instance, one could consider domination and connected domination in a generalized grid (box) graph, such as a Cartesian product of $n$ paths on $p$ vertices. This graph on $p^{n}$ vertices is the hypercube when $p=2$. Perhaps the more natural graph to study is a product of $n$ cycles on $p$ vertices. Note that for $p=4$ it is the same graph as $Q_{2n}$. Edenfield [7] recently studied products of cycles and products of complete graphs, both generalizations of the hypergraph questions in this paper.

Joshua Cooper suggests considering powers of graphs. That is, for a graph $G=(V,E)$, such as the hypercube, fix integer $r>0$ and consider the same questions as before, but for the graph $G^{r}$: This graph also has vertex set $V$, but now edges join vertices at distance at most $r$ in $G$. This is motivated by covering codes of radius $r$.

8 Acknowledgements

We express gratitude to Sergei Bezrukov for sharing his questions and notes that led to this study. Particular thanks go to colleague Joshua Cooper for bringing to our attention the key domination result $\gamma(Q_{n})\sim 2^{n}/n$ as presented in [5].

We appreciate support for travel to Taiwan to work on this project from Mathematics Research Promotion Center Grant 108-17 and Ministry of Science and Technology Grant 107-2115-M-005-002-MY2.