ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 14 (2018) 387-395 https://doi.org/10.26493/1855-3974.1172.bae
(Also available at http://amc-journal.eu)
On domination-type invariants of Fibonacci cubes and hypercubes*
Jernej Azarija
Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia
Sandi KlavZar t
Faculty of Mathematics and Physics, University of Ljubljana, Slovenia Faculty of Natural Sciences and Mathematics, University ofMaribor, Slovenia Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia
Yoomi Rho Seungbo Sim
Department of Mathematics, Incheon National University, Korea
Received 6 August 2016, accepted 11 August 2017, published online 28 October 2017
The Fibonacci cube rn is the subgraph of the n-dimensional cube Qn induced by the vertices that contain no two consecutive 1s. Using integer linear programming, exact values are obtained for Yi(Tn), n < 12. Consequently, 7t(rn) < 2Fn_10 + 21Fn-8 holds for n > 11, where Fn are the Fibonacci numbers. It is proved that if n > 9, then Yt(rn) > |-(Fn+2 - 11)/(n — 3)] - 1. Using integer linear programming exact values for the 2-packing number, connected domination number, paired domination number, and signed domination number of small Fibonacci cubes and hypercubes are obtained. A conjecture on the total domination number of hypercubes asserting that yt(Qn) = 2n-2 holds for n > 6 is also disproved in several ways.
Keywords: Total domination number, Fibonacci cube, hypercube, integer linear programming, covering codes.
Math. Subj. Class.: 05C69, 68R10, 94B05
*We are grateful to an anonymous referee for a very careful reading on the manuscript. t Author to whom correspondence should be addressed.
* On 2 June 2017, Yoomi Rho tragically passed away in the middle of her scientific career. E-mail addresses: jernej.azarija@gmail.com (Jernej Azarija), sandi.klavzar@fmf.uni-lj.si (Sandi KlavZar), tmdqh7507@naver.com (Seungbo Sim)
Abstract
©® This work is licensed under http://creativecommons.Org/licenses/by/3.0/
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1 Introduction
Fibonacci cubes were introduced by Hsu [19] because of their appealing properties applicable to interconnection networks. Afterwards they have been extensively studied and found additional applications, see the survey [23]. The interest for Fibonacci cubes continues, recent research of them includes asymptotic properties [24], connectivity issues [7], the structure of their disjoint induced hypercubes [14, 30], the (non)-existence of perfect codes [5], and the q-cube enumerator polynomial [31]. From the algorithmic point of view, Ramras [29] investigated congestion-free routing of linear permutations on Fibonacci cubes, while Vesel [34] designed a linear time recognition algorithm for this class of graphs.
The domination number of Fibonacci cubes was investigated by now in two papers. Pike and Zou [28, Theorem 3.2] proved that y(T„) > \(Fn+2 - 2)/(n - 2)] for n > 9, where Fn are the Fibonacci numbers: F0 = 0, F1 = 1, Fn = Fn-1 + Fn-2 for n > 2. Exact values of Y(rn) for n < 8 were also obtained in [28]. In the second related paper [9] the domination number of Fibonacci cubes was then compared with the domination number of Lucas cubes.
In this note we turn our attention to domination invariants of Fibonacci cubes and of hypercubes with a prime interest on the total domination. We proceed as follows. In the rest of this section we introduce concepts and notation needed. Then, in Section 2, we determine the exact value of the total domination number of rn for n < 12, and obtain an upper bound and a lower bound on jt (rn). In Section 3 we use integer linear programming to either extend or obtain values for several domination-type invariants on Fibonacci cubes and hypercubes. In the final section we consider the total domination of hypercubes with respect to a recent conjecture from [22]. In particular, using known results from coding theory we show that the conjecture does not hold. It is also observed that for any c > 0 there exists n0 G N, such that if n > n0, then 7t(Qn) < 2n-c.
The n-dimensional (hyper)cube Qn, n > 1, is the graph with V(Qn) = {0,1}n, two vertices being adjacent if they differ in a single coordinate. For convenience we also set Q0 = K1. The vertices of Qn will be briefly written as binary strings b1... bn. A Fibonacci string of length n is a binary string b1... bn with b • bi+1 = 0 for 1 < i < n. Fibonacci strings are thus binary strings that contain no consecutive 1s. The Fibonacci cube rn, n > 1, is the subgraph of Qn induced by the Fibonacci strings of length n. It is well known that |V (rn) | = Fn+2.
If u is a binary string, then the number of its bits equal to 1 is the weight of u. If u and v are binary strings, then uv denotes the usual concatenation of the two strings. If u is a binary string and X a set of binary strings, then uX = {ux : x G X}.
Let G be a graph. Then D C V(G) is a dominating set if every vertex from V(G) \ D is adjacent to some vertex from D. The domination number 7(G) is the minimum cardinality of a dominating set of G. D is a total dominating set if every vertex from V(G) is adjacent to some vertex from D. The total domination number Yt(G) is the minimum cardinality of a total dominating set of G. Note that the total domination number is not defined for graphs that contain isolated vertices, hence unless stated otherwise, all graphs in this paper are isolate-free. For more information on the total domination in graphs see the recent book [17] and papers [11, 12].
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2 Total domination in Fibonacci cubes
In this section we present exact values of 7t(rn) for n < 12, prove an upper bound on 7t(rn), and a lower bound on Yt(rn). The exact values were obtained by computer and are collected in Table 1, where the order of the cubes is also given so that the complexity of the problem is emphasized. In particular, |V(ri2) | = 377.
Table 1: Exact total domination numbers of Fibonacci cubes up to dimension 12.
n	1	2	3	4	5	6	7	8	9	10	11	12
|V (r„)|	2	3	5	8	13	21	34	55	89	144	233	377
Yt(rn)	2	2	2	3	5	7	10	13	20	30	44	65
More precisely, the results from Table 1 were obtained using integer linear programming as follows. Suppose we associate to each vertex v e V(T„) a binary variable xv. The problem of determining Yt(rn) can then be expressed as a problem of minimizing the objective function
Xv,
vev (r„)
subject to the condition that for every v e V (T„) we have
> i.
The value of the objective function is then jt (T„).
We have found out that the most efficient solver for the above problem is Gurobi™ Optimizer [15]. For example, it takes less than 9s to compute Yt(r12) on a standard desktop machine. On the other hand, we were not able to make the computation for 7t(T13) in real time (note that the order of r13 is 610), we could only get the estimates
97 < 7t(ris) < 101.
Using the above computations, the following result can be derived.
Theorem 2.1. If n > 11, then Yt(r„) < 2F„_io + 21F„_g.
Proof. Consider the so-called fundamental decomposition of rn into the subgraphs induced by the vertices that start with 0 and 10, respectively (cf. [23]). These subgraphs are isomorphic to rn-1 and rn_2 respectively, hence we infer that Yt(rn) < Yt(rn-1) + 7t(rn_2). From the above computations we know that Yt(r11) = 44 and Yt(r12) = 65. Define the sequence (an), n > 11, with a11 = 44, a12 = 65, and an = an_1 + an_2 for n > 13. Then one can check by a simple induction argument that an = 2Fn_10 + 21Fn_8 holds for any n > 11. Since Yt(rn) < an the argument is complete.	□
Arnautov [3] and independently Payan [27] proved that
Y(G)	£ 1	(2.1)
S +1 j
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holds for any graph G of minimum degree J. Since J(rn) = |(n + 2)/3j, cf. [25, Corollary 3.5], and because Yt < 27, we get that
L nf5 J
Yt(rn) < 2iif T j.	(2.2)
L 3 J j=i j
Computing the values of the right-hand side of the bound of Theorem 2.1 and of (2.2) we find out that Theorem 2.1 is better than the bound of (2.2) for n < 33.
By using the fact 7t(r13) < 101 that was obtained by our computations, the bound of Theorem 2.1 can be further improved to give
Yt(r„) < 601Fn_i - 371F„, n > 12.
We continue by establishing a lower bound on Yt (rn).
Theorem 2.2. If n > 9, then
7t(r„) >
Fn+2 - 11 n3
1.
Proof. The proof mimics the proof of [28, Theorem 3.2] which gives a lower bound on the domination number of Fibonacci cubes, hence we will not give all the details.
For a graph G and its total dominating set D we introduce the over-total-domination of D in G as ODG(D) =	deg(v) - |V(G)|. Consider now rn, n > 9, and let D be a
total dominating set of rn. In rn, the vertex 0n is the unique vertex of degree n, vertices 10n-1 and 0n-11 have degree n - 1, and all other vertices of weight 1 have degree n - 2. In addition, the vertices 1010n-3,10n-21, and 0n-3101 are of degree n - 2, while all other vertices of rn have degree at most n - 3, cf. [25].
Let k be the number of vertices of weight 1 from D \ {10n-1,0n-11}. In addition, let i = |D n {1010n-3,10n-21,0n-3101}|. Note that k + i is the number of vertices from D that have degree n - 2. The proof now proceeds by considering the cases that happen based on the membership of the vertices 0n, 10n-1, and 0n-11 in D. Here we consider only the case when {0n, 10n-1, 0n-11} C D. We have:
ODg(D) < n + 2(n - 1) + (k + i)(n - 2) + (7t(rn) - 3 - k - i)(n - 3) - F„+2 .
Since clearly ODG(D) > 0, from the above inequality we derive that Yt (rn)(n - 3) > Fn+2 - k - i - 7. Because k + i < n +1 we get
, . Fn+2 - k - i - 7 ^ F„+2 - (n +1) - 7 7i(in) > -o- >
n - 3	n - 3
Fn+2 - 11 - (n - 3) Fn+2 - 11
1
n - 3	n - 3
and the stated inequality holds in this case. All the other cases are treated similarly. □
We conclude the section with Table 2 in which known values and current best bounds on 7i(r„) for n < 33 are collected. The values for n < 12 were computed using the linear program explained above. The bounds for Yt(r13) were established by Gurobi, and we conjecture that in fact rt(r13) = 101. Finally, the remaining bound in Table 2 were obtained by the bounds given in Theorems 2.1 and 2.2. Recall that n = 33 is the last value for which Theorem 2.1 gives a better bound than the bound (2.2).
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Table 2: Exact values and current best bounds on Yi(Tn), n < 33.
n	Yt(r„)	n	Yt(r„)	n	Yt(r„)
1	2	12	65	23	3749-13276
2	2	13	97-101	24	5779-21481
3	2	14	87-174	25	8926-34757
4	3	15	131-283	26	13816-56238
5	5	16	196-457	27	21424-90995
6	7	17	296-740	28	33280-147233
7	10	18	449-1197	29	51778-238228
8	13	19	682-1937	30	80676-385461
9	20	20	1040-3134	31	125876-623689
10	30	21	1590-5071	32	196649-1009150
11	44	22	2438-8205	33	307580-1632839
3 Additional invariants on small Fibonacci cubes and hypercubes
The integer linear programming approach can be used to compute several additional invariants of Fibonacci cubes (and other graphs). This has recently been done by Ilic and Milosevic in [20], where they have computed the domination number, the 2-packing number, and the independent domination number of low dimensional Fibonacci cubes. In particular, they have used integer linear programming to confirm the conjecture from [9] stating that Y(r9) = 17. In addition, an integer linear programming model for the connected domination number has been presented in [13]. In this section we add to the list of integer linear programming models paired domination and signed domination. The concepts mentioned in this paragraph that have not been introduced yet are defined next.
A set X C V(G) is a 2-packing if d(x,y) > 3 holds for any x, y G X, x = y. The maximum size of a 2-packing of G is the 2-packing number of G denoted p(G). The independence domination number i(G) of G is the minimum size of a dominating set that induces no edges [26]. The connected domination number 7c(G) of G is the order of a smallest dominating set that induces a connected graph [10]. The paired domination number 7p(G) is the order of a smallest dominating set S C V(G) such that the graph induced by S contains a perfect matching [2]. Finally, we say that f: V(G) ^ {-1,1} is a signed dominating function if	[v] f (u) > 1 holds for every v G V(G), where
N[v] is the closed neighborhood of v, that is, N[v] = {v} U {u : vu G E(G)}. The signed domination number 7s(G) is the minimum of J2vev(G) f (v) taken over all signed dominating functions f of G, see [18].
We now present the problems to determine the paired domination number of a graph and the signed domination number of a graph as integer linear programs. To model the paired domination problem for a graph G we introduce a binary variable xe indicating whether the edge e G E(G) is present in the graph induced by a paired dominating set of G. Then we can model the problem as follows:
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Y1 xe
eEE(G)
^ xuv < 1, v e V(G)
u^v
^^ Xuw > 1, v e V(G) .
u^v w^u
Similarly, to model the signed domination number we introduce a binary variable xv associated with every vertex v e V(G) indicating whether v is assigned weight 1 or —1, respectively. Then we have the following linear program.
minimize	(2xv — 1)
vev (G)
subject to ^ (2xu — 1) > 1, v e V(G).
ueN [v]
Our computational results are collected in Tables 3 and 4. In the rows for Y(rn), p(rn), and i(rn), the results from [20] are in normal font, while the new values are in bold. We have thus extended the results from [20] for one additional dimension. It is interesting to observe that the gap between the independent domination number and the domination in dimension 9 is equal to 2, but then in dimensions 10 and 11 the difference goes down to 1.
Table 3: Additional invariants for small Fibonacci cubes and hypercubes.
n	1	2	3	4	5	6	7	8	9	10	11	12
Y(r„)	1	1	2	3	4	5	8	12	17	25	39	54-61
p(r„)	1	1	2	2	3	5	6	9	14	20	29	42
i(r„)	1	1	2	3	4	5	8	12	19	26	40	?-?
minimize subject to
Table 4: Additional invariants for small Fibonacci cubes and hypercubes.
n	1	2	3	4	5	6	7	8	9	10
7c(r„)	1	1	2	3	5	7	10	14	22	
Yc(Qn)	1	2	4	6	10	16	28			
Yp(r„)	2	2	2	4	6	8	10	14	20	30
Yp ( Qn )	2	2	4	4	8	14	24	32		
Ys(r„)	2	3	3	2	5	9	10	17	25	40
Ys(Qn)	2	2	4	6	12	16	32			
4 On total domination in hypercubes
It has recently been conjectured in [22, Conjecture 4.6] that Yt(Qn) = 2n-2 holds for n > 6. In [4] Arumugam and Kala first observed that 7t(Qi) = 7t(Q2) = 2 and Yt (Q3 ) = 7i (Q4 ) =4, and then followed by proving that jt (Q5 ) = 8 [4, Theorem 5.1] and
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7t(Q6) = 14 [4, Theorem 5.2]. The last result is then a sporadic counterexample to the conjecture. Actually, at this moment the exact value of 7t(Qn) is known for n < 10: 7t(Q7) = 24, 7t(Qs) = 32, 7t(Qg) = 64, and 7t(Q10) = 124, see [33, Appendix B, p. 40]. Hence Q7 and Q10 are additional sporadic counterexamples (and so are Q8 and Qg since Yt(Qs) = 32 = 26 and 7t(Qg) = 64 = 27).
Total dominating sets of Qn can be in coding theory equivalently described as covering codes of empty spheres (of length n and covering radius 1). The following result was first proved back in [21], see also [35, Theorem 1(b)]. Let us rephrase the result here in graph-theoretical terms and give a corresponding argument.
Proposition 4.1. If n = 2k, k > 0, then 7t(Qn) = 2n-k.
Proof. From [32] we know that if n = 2k, then 7(Qn) = 2n-k and from [16] that if n = 2k - 1, then also 7(Qn) = 2n-k. Let n = 2k and consider Qn. Let QL-1 and QR-1 be the subgraphs of Qn induced by the sets of vertices X0 = {0b2... bn : b € {0,1}} and X1 = {1b2... bn : bj € {0,1}}, respectively. Clearly, V(Qn) partitions into X0 and X1, and in Qn every vertex of X0 has a unique neighbor in X1. Moreover, Q^_1 and QR-1 are both isomorphic to Qn-1. Let CL be a perfect code of Q^_1 and let CR be its copy in QR-1. Then CL U CR is a total dominating set of Qn of order 2n-k. Since on the other hand 7t(Qn) > 7(Qn) = 2n-k, the conclusion follows.	□
It follows from (2.1) that
-,(G) < |V(G)| (i±|+±i!)	(4.1)
holds for any graph G. Hence, again using the fact that 7t(G) < 27(G), we get for hypercubes that
„«,.) < 2n+. (i±M) .
Directly from this inequality we infer:
Remark 4.2. For any c > 0 there exists n0 € N, such that if n > n0, then
Yt(Qn) < 2n-c .
Two remarks are in place here. First, (4.1) also follows from a more general result on transversals in hypergraphs due to Alon [1]. Second, the state of the art on the upper bounds on the domination number in terms of the minimum degree and the order of a given graph is given in [8].
It follows from the fact that 7t(Qn) < 27(Qn-1) and from Proposition 4.1 that
Yt(Q2fc+1) < 27(Q2k) = 22 -k+1. As proved in [32], the equality actually holds here, that is, 7t(Q2k+1) = 22 -k+1. More generally, 7t(Qn+1) = 27(Qn) holds for any n, a result very recently proved in [6].
Acknowledgment
The authors acknowledge the financial support from the Slovenian Research Agency (research code funding No. P1-0297) and from the Basic Science Research Program through
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the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology grant 2011-0025319. Supported also by the bilateral Korean-Slovenian project BI-KR/13-14-005 and the International Research & Development Program of the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (MSIP) of Korea (Grant number: NRF-2013K1A3A1A15003503).
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