/^creative ^commor
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 16 (2019) 245-255 https://doi.org/10.26493/1855-3974.1591.92e
(Also available at http://amc-journal.eu)
ARS MATHEMATICA CONTEMPORANEA
On the domination number and the total domination number of Fibonacci cubes
Elif Saygi *
Department of Mathematics and Science Education, Hacettepe University, 06800, Beytepe, Ankara, Turkey
Received 2 February 2018, accepted 9 October 2018, published online 4 January 2019
Abstract
Fibonacci cubes are special subgraphs of the hypercube graphs. Their domination numbers and total domination numbers are obtained for some small dimensions by integer linear programming. For larger dimensions upper and lower bounds on these numbers are given. In this paper, we present the up-down degree polynomials for Fibonacci cubes containing the degree information of all vertices in more detail. Using these polynomials we define optimization problems whose solutions give better lower bounds on the domination numbers and total domination numbers of Fibonacci cubes. Furthermore, we present better upper bounds on these numbers.
Keywords: Fibonacci cubes, domination number, total domination number, integer linear programming.
Math. Subj. Class.: 05C69, 68R10, 11B39
1 Introduction
Let G = (V(G), E(G)) be a graph with vertex set V(G) and edge set E(G). D C V(G) is a dominating set of G if every vertex in V(G) either belongs to D or is adjacent to some vertex in D. The domination number y(G) is defined as the minimum cardinality of a dominating set of the graph G. Similarly, D C V(G) is a total dominating set if every vertex in V(G) is adjacent to some vertex in D and the total domination number jt(G) is defined as the minimum cardinality of a total dominating set of G. Note that the total domination number is defined for isolate-free graphs and it is not defined for the graphs that contain isolated vertices. The domination number of the Fibonacci cubes rn is first given
* Supported by TUBiTAK under Grant No. 117R032. The author would like to thank the anonymous reviewers for their valuable comments and would like to thank Prof. Dr. Omer Egecioglu for useful discussions and suggestions.
E-mail address: esaygi@hacettepe.edu.tr (Elif Saygi)
©® This work is licensed under https://creativecommons.org/licenses/by/4.0/
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in [14] and [2]. These results are extended in [8] by using integer linear programming for some cases. Total domination number of rn is considered in [1], in which an upper bound and a lower bound on Yt(rn) are obtained. The exact values of y(T„) and Yt(rn) are also considered by using integer programming in [1]. The upper bound on Yt(rn) given in [1] is improved in [15]. We summarize these results in Section 2. The aim of this work is to improve some of the results given in [1] and [15].
The hypercube Qn of dimension n > 1 is the graph with vertex set V(Qn) = {0,1}n, in which two vertices are adjacent if they differ in one coordinate. For convenience Q0 = K1. All the vertices of Qn are labeled by the binary strings of length n. The Fibonacci cubes rn are special subgraphs of Qn and they were introduced by Hsu [7] as a model of interconnection networks. In literature, many interesting properties of the Fibonacci cubes have been investigated, see survey [9] for details. In recent years results on disjoint hypercubes in rn are presented in [5, 13, 16] and the cube enumerator polynomial of rn is considered in [10, 11, 17] and many combinatorial results are given. The domination-type invariants of rn are considered in [1, 2, 8, 14, 15] and some numerical results and bounds are presented.
It is known that Fibonacci strings of length n are the binary strings of length n that contain no consecutive ones. For this reason we can write
V (rn) = {b1b2 ••• bn | bi e {0,1}, 1 < i < n, and b, • bi+1 =0 for 1 < i < n} and
E (rn) = {(u,v) | U,v e V (rn), dH (u, v) = 1},
where dH(u, v) denotes the Hamming distance between u and v, that is, the number of different coordinates in u and v. The number of vertices of the Fibonacci cubes rn is Fn+2, where Fn are the Fibonacci numbers defined as F0 =0, F1 = 1 and Fn = Fn—1+Fn—2 for n > 2. For n > 2 we will use the following formulation for the fundamental decomposition of rn (see, [9]):
rn
orn_ i + iorn_ 2.
(1.1)
Here note that 0rn—1 is the subgraph of rn induced by the vertices that start with 0 and 10rn-2 is the subgraph of rn induced by the vertices that start with 10. Furthermore, 0rn-1 has a subgraph isomorphic to 00rn—2, and there is a matching between 00rn—2 and 10rn —2 (see Figure 1).
Figure 1: Fundamental decompositions of the Fibonacci cube rn, n > 3.
In this paper, we present upper bounds on Y(Tn) and Yî(T„). Furthermore, we introduce the up-down degree polynomials for rn containing the degree information of all vertices V (T„) in more detail. Using these polynomials we define optimization problems whose solutions give lower bound on y (rn) and Yt(rn).
E. Saygi: On the domination number and the total domination number of Fibonacci cubes 247
2 Known results and new upper bounds on 7(rn) and Yt(rn)
In this section, first we summarize some known results on the domination number and the total domination number of Fibonacci cubes and then we present new uppper bounds for these numbers. We start with Figure 2 and Figure 3 showing a dominating set and a total dominating set for small dimensional rn's.
Figure 2: r0,..., r5 and their dominating sets.
Figure 3: ri,..., r5 and their total dominating sets.
We collect the known values of y(T„) and Yi(T„) in Table 1. The values of 7(T„) for n < 8 are obtained in [14]. The other values of y(T„ ) are obtained by integer programming. For n = 9 and n = 10 they are obtained in [8] and for n = 11 and n =12 they are obtained in [1]. Similarly, all the values of Yi(T„) given in Table 1 are obtained by computer using integer programming in [1].
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Table 1: Known values of 7(T„) and 7t(rn).
n	1	2	3	4	5	6	7	8	9	10	11	12	13
|V (r„)|	2	3	5	8	13	21	34	55	89	144	233	377	610
Y(r„)	1	1	2	3	4	5	8	12	17	25	39	54 - 61	
7t(r„)	2	2	2	3	5	7	10	13	20	30	44	65	97 -101
Now we describe the integer linear programming used in [8] and [1]. Suppose each vertex v e V(rn) is associated with a binary variable xv. Let N(v) be the set of vertices adjacent to v and N[v] = N(v) U {v}. The problems of determining Y(rn) and Yt(rn) can be expressed as a problem of minimizing the objective function
E ^
vev (r„)
(2.1)
subject to the condition that for every v e V(rn) we have
xa > 1 (for domination number),
aeN [v]
xa > 1 (for total domination number).
aeN (v)
The value of the objective function is then 7(rn) and Yt(rn) respectively. Note that this problem has Fn+2 variables and Fn+2 constraints. In [1], it is stated that Y(r12) and 7i(r13) were not computed in real time using the above optimization problem. They got the estimates
54 < Y(r12) < 61 and 97 < 7t(r13) < 101.
Here, the main difficulty is the order of rn which equals to the number of variables and the number of constraints.
By using the degree information of the vertices in rn the following lower bound on 7 (rn) is presented in [14].
Theorem 2.1 ([14]). If n > 9, then
Y(r„) >
Fn+2 — 2 n - 2
By using a similar technique the following lower bound on jt (rn) is obtained in [1]. Theorem 2.2 ([1]). If n > 9, then
~ Fn+2 - 11"
7t(r„) >
n — 3
- 1.
In Section 3 we propose an optimization problem having less number of variables and constraints to estimate lower bounds on Y(rn) and Yt(rn). Our results improve the lower
v
E. Saygi: On the domination number and the total domination number of Fibonacci cubes 249
bounds given in Theorem 2.1 and Theorem 2.2 and we present some numerical values in Table 2 and Table 3.
By using the exact values in Table 1 and the fundamental decomposition (1.1) of rn, the following upper bound on Yi(Tn) is obtained in [1].
Theorem 2.3 ([1]). If n > 11, then Yt(T„) < 21F„_g + 2F„_io.
In [1], using the computer result Yt(r13) < 101 the upper bound in Theorem 2.3 improved to
Yt(rn) < 601K-1 - 371 F„, n > 12.
These two upper bounds further improved in [15] by using the values of y(rn) and the fundamental decomposition (1.1) of rn more than once.
Theorem 2.4 ([15]). If n > 15, then
Y(r„) < Yt(rn) < 3Y(r„-3) + 2Y(r„_4)
< 116F„ - 187K-1 = 21F„-s - (2K-10 + Fn_12).
Furthermore, Yt(r14) < 166 .
We implemented the same integer linear programming problem (2.1) using CPLEX in NEOS Server [3, 4, 6] for n =13 and obtain the estimates (takes approximately 2 hours)
78 < Y(r13) < 93.
Using this result with y (r12) < 61 we obtain the following bound on the domination number of rn.
Theorem 2.5. If n > 12, then Y(rn) < 21Fn-8 - (2Fn-10 + 8Fn-12).
Proof. The proof mimics the proof of [1, Theorem 2.1]. By the fundamental decomposition (1.1) of r„ we have y(r„) < y(r„-1) + Y(r„-2). We know that y(^2) < 61 and Y(r13) < 93. For n > 12 define the sequence (bn) with bn = 6n-1 + 6„_2 where b12 = 61 and b13 = 93. Then by induction we have bn = 21Fn-8 - 2Fn-10 - 8Fn-12 for any n > 12. We complete the proof since y(rn) < bn for n > 12.	□
Combining the results in Theorem 2.5 and Theorem 2.4 we get the following result which improves Theorem 2.3 and Theorem 2.4.
Theorem 2.6. If n > 16, then
Yt(rn) < 21F„_8 - (2K-10 + 8F„-12).
3 Up-down degree enumerator polynomial
In this section we present the up-down degree enumerator polynomial for rn. It contains the degree information of all vertices V (T„) in more detail. Using this polynomial we write optimization problems whose solutions are lower bounds on Y(rn) and 7t(rn).
For each fixed v G V(rn) we write a monomial where d = w(v) is the Hamming weight of v and u is deg(v) - d (that is, deg(v) = u + d). Recall that, by the definition of rn, (v, v') G E(rn) if and only if dH (v, v') = 1. Therefore, the number of neighbors of v
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whose weight is one more than the weight of v (say up neighbors of v, w(v') = w(v) + 1) is u and the number of neighbors of v whose weight is one less than the weight of v (say down neighbors of v, w(v') = w(v) — 1) is d. For this reason we call the polynomial
Pn(x,y) = ^ xdeg(v)-w(v)yw(v) = ^ xuyd vev (r„)	vev (r„)
as the up-down degree enumerator polynomial of rn. By using the fundamental decomposition (1.1) of rn we obtain the following recursive relation which will be useful to calculate Pn(x, y).
Theorem 3.1. Let Pn(x,y) be the up-down degree enumerator polynomial of rn. Then for n > 3 we have
Pn(x, y) = xP„-i(x, y) + yPn-2(x,y) + yP„_3(x,y) — xyP„_3(x,y),
where
Po(x,y) = 1, Pi(x,y)= x + y and P2(x, y) = x2 + 2y.
Proof. P0, P1 and P2 are clear from Figure 2. Assume that n > 3. We know that the up-down degree enumerator polynomials of rn-1, rn-2 and rn-3 are Pn-1(x, y), Pn-2(x, y) and Pn_3(x, y) respectively. By (1.1) we have
r„ = or„_i + ior„_2	(3.1)
= (oor„_2 + oior„_3) + iorn_2	(3.2)
and there is a matching between oorn-2 and iorn-2 (see also Figure 1). From this decomposition we have the following three different cases:
1.	Assume that v g iorn-2. These vertices are the ones in rn-2 whose weights d = w(v) increase by one in rn. Furthermore, their degrees increase by one due to the matching between oorn-2 and iorn-2, which means that u = deg(v) — w(v) remains the same in rn. Therefore, these vertices contribute yPn-2(x, y) to Pn(x, y).
2.	Assume that v g oiorn-3. These vertices are the ones in rn-3 whose weights d = w(v) increase by one in rn and their degrees increase by one due to the matching between oiorn-3 and ooorn-3 c oorn-2, which means that u = deg(v)—w(v) remains the same in rn. Therefore, these vertices contribute yPn-3(x, y) to Pn(x, y).
3.	Assume that v g oorn-2. These vertices are the ones in orn-1 that are not in oior„_3. In rn -i the up-down degree enumerator polynomial of these vertices becomes Pn-1(x, y) — yPn-3(x, y). The weights d = w(v) of all such vertices remain the same in rn but their degrees increase by one due to the matching between oorn-2 and iorn-2, that is, u = deg(v) — w(v) increase by one in rn. Therefore, these vertices contribute x(Pn-1(x, y) — yPn-3(x, y)) to Pn(x, y).
By adding all of the above contributions we get the desired result.	□
Now we describe an optimization problem using the up-down degree enumerator polynomial Pn(x, y). Let be a total dominating set of rn. Then by the definition of Fibonacci cubes for every vertex v g V(rn) with weight w(v) then there must exist a vertex
E. Saygi: On the domination number and the total domination number of Fibonacci cubes 251
vD € N(v) n DT with w(vD) = w(v) + 1. Furthermore, assume that for any fixed vertex vD € Dt its corresponding monomial be xUyd in the Pn(x, y). This means that vD dominates u distinct vertices v € V(rn) with weight w(v) = w(vD) + 1 and d distinct vertices v € V(rn) with weight w(v) = w(vD) - 1. Note that for all vD € DT some of the dominated vertices may coincide. Now assume that
Pn(x,y)= E xuyd = E cUx"yd.	(3.3)
vev (r„)
For each pair (u, d) in Pn(x, y) we associate an integer variable zU which counts the number of vertices in DT with weight d and degree u + d, that is, the number of vertices in DT having d down neighbors and u up neighbors. Clearly, we have the bounds 0 < zU < cU. Our aim is to minimize |DT |, that is, our objective function is to minimize
E zu.
U,d
Then by the above observation to dominate all the vertices having a fixed weight d such that 1 < d < [ n 1 - 1 we must have
rd : E(u ■ zU-i + (d + 1) ■ zU+1) > E cU
d
since any vertex with weight d - 1 having u up neighbors can dominate u distinct vertices with weight d and any vertex with weight d +1 (all have d +1 down neighbors) can dominate d +1 distinct vertices with weight d. By the same argument, for d = 0 we must have
ro : E zU >E cU = 1
and for d = [ n 1 we must have
rr21 : Eu ' zU^l-1 ^Ecrf 1 \ n + 1 if n is even.
1	if n is odd,
n
2
Now subject to these constraints r0,..., rp n 1 the value of the objective function will be a lower bound on Yt(rn). Similarly, to find a lower bound on Y(rn) we need to modify all of the constraints rd, 0 < d < [n 1. By the definition of the dominating set, for each fixed d we need to add all of the variables zU to the left side of the constraint rd.
Remark 3.2. We remark that using [12, Theorem 4.6] we can easily obtain the coefficients cU in (3.3). By the definition of the up-down degree enumerator polynomial we know that cU is the number of vertices in rn whose number of up neighbors is u and weight is d. That is, cU equals to the number of vertices of rn having degree u + d and weight d. Therefore, [12, Theorem 4.6] gives
d +1 \fn - 2d' n — 2d — u + 1/1 u
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Remark 3.3. We know that the number of vertices of rn with weight d is equal to the right hand side of the above constraints rd. By definition of rn this number is equal to the number of Fibonacci strings of length n and weight d. Therefore we have
n — d +1
"V d
u	v
Remark 3.4. To find the number of variables zU we need to find the number of monomials in Pn(x, y). Assume that n is even. Then by the structure of the vertices in Fibonacci cubes (it can also be seen from the structure of Fibonacci strings) n - 3d < u < n - 2d. Therefore the number of variables zU becomes
L fJ	f
^(d +1)+ ^ (n - 2d +1)
d=0	d=LfJ+1
which is equal to
s2 - 2sr +3r(r + 1) +1
where r = [f J and s = n/2. Similarly, if n is odd we obtain that the number of variables is
s2 — s(2r + 1) + r(3r2+-5) + 2 where r = [ f J and s = [ f ].
Now we illustrate our optimization problem for n = 14. We have the following polynomial by Theorem 3.1.
PM(x,y) = 8y7 +
7y6x2 + 42y6x + 35y6 +
6y5x4 + 60y5x3 + 120y5x2 + 60y5x + 6y5 +
5y4x6 + 60y4x5 + 150y4x4 + 100y4x3 + 15y4x2 +
4y3x8 + 48y3x7 + 112y3x6 + 56y3x5 +
3y2x10 + 30y2 x9 + 45y2x8 +
2yx12 + 12yx11 +
„14
and this polynomial corresponds to the following optimization problem: Objective function:
min : zl4 + z12 + zl1 + z210 + zf + zf + zf + zf + zf + zf +
z46 + z4 + z4 + z4 + z2 + z54 + zf + z52 + zl + z50 + z2 + zl + z0 + z0
E. Saygi: On the domination number and the total domination number of Fibonacci cubes 253
Constraints for jt (r 14):
rr :		2z	2 + z1 6 + z6	>8
r6 :		4zf + 3zf + 2z2 + zf	+ 7z0	> 84
r5 :	6z46	+ 5z| + 4z| + 3z| + 2z| + 6z| + 6z£	+ 6z0	> 252
r4 :	8zf + 7z7	+ 6z| + 5zf + 5zf + 5zf + 5zf + 5zf	+ 5zf	> 330
r3:	10z10	+ 9z2 + 8zf + 4z| + 4zf + 4z| + 4z|	+ 4z|	> 220
r2 :		12z12 + Hz111 + 3zf + 3z7 + 3z|	+ 3zf	> 78
r1 :		14z14 + 2z10 + 2z9	+ 2zf	> 14
r0 :		z12 z1	+ z111	>1
Constraints for 7 (r 14):
rr: r6: r5 :
r4 :
r3:
r2 : r1 :
ro:
2z2 + + z° > 8 4zf + 3zf + 2zf + Z51 + z2 + + z0 + 7z0 > 84 6z6 + 5z4 + 4z4 + 3z4 + 2z4 + z4 + zf +
z5 + z1 + z50 + 6z2 + 6zg + 6z0 > 252
8zf + 7zr + 6z6 + 5zf + z6 + z4 + z4 + z| +
z2 + 5z4 + 5zf + 5zf + 5z1 + 5z0 > 330
10z210 + 9z9 + 8zf + zf + z7 + zf +
zf + 4z6 + 4z4 + 4z4 + 4zf + 4z4 > 220 12z12 + llz!1 + z10 + z29 + zf + 3zf + 3zf + 3z6 + 3zf > 78 14z14 + z12 + z!1 + 2z10 + 2z9 + 2zf > 14 z14 + z112 + z111 > 1
Bounds:
z14	< 1	z12	<2	z111	< 12	z210	<3	z29	< 30
zf	< 45	zf	<4	z7	< 48	z6	< 112	zff	< 56
z46	<5	z4	< 60	z44	< 150	zf	< 100	z42	< 15
z4	<6	zf	< 60	z52	< 120	zf	< 60	zf	<6
z62	<7	z61	< 42	z0	< 35	z0	<8		
The value of the objective function gives a lower bound on Y(r14) and Yt(r14) respectively. Note that the above problem have only 24 variables and 8 constraints (instead of having 987 variables and 987 constraints, see Section 2). In general using the up-down degree enumerator polynomial Pn(x, y) of rn in Theorem 3.1 we can write an optimization problem having less number of variables z^ (see Remark 3.4) and [f 1 + 1 constraints rd. The solutions of these problem give lower bounds on 7 (rn) and Yt(rn). One can easily see that the number of variables and the number of constraints are very smaller than the ones in the optimization problem described in Section 2.
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For illustration we implemented the above integer linear programming problem using CPLEX in NEOS Server [3, 4, 6] for 13 < n < 26 and immediately obtain the lower bounds presented in Table 2 and Table 3. Note that for n = 26, the number of variables in our optimization problem is 70 by Remark 3.4 and it is F28 = 317811 for the general optimization problem (2.1). In addition, the upper bounds in these tables are obtained by Theorem 2.5 for n > 14 and Theorem 2.6 for n > 16. Note that the first bounds in both tables are obtained in [1] and the upper bounds on Yt(ri4) and Yt(ri5) comes from Theorem 2.4.
Table 2: Current best bounds on Y(rn), 12 < n < 26.
n	Y(r„)	n	Y(r„)	n	Y(r„)
12	54 - 61	17	344 - 648	22	3060 - 7189
13	78 - 93	18	528 -1049	23	4748 -11632
14	98 -154	19	819 -1697	24	7381 -18821
15	148 - 247	20	1270 - 2746	25	11472 - 30453
16	224 - 401	21	1970 - 4443	26	17912 - 49274
Table 3: Current best bounds on Yt(rn), 13 < n < 26.
n	Y(r„)	n	Y(r„)	n	Y(rn)
13	97 -101	18	578 -1049	23	5075 -11632
14	110 -166	19	890 -1697	24	7865 -18821
15	164 - 261	20	1374 - 2746	25	12191 - 30453
16	246 - 401	21	2121 - 4443	26	19033 - 49274
17	376 - 648	22	3281 - 7189		
Remark 3.5. For n = 12, Theorem 2.1 gives Y(r12) > 38 and Theorem 2.2 gives 7i(r12) > 40. The values of the objective function in our optimization problems having 19 variables and 7 constraints give lower bounds Y(r12) > 44 and 7t(r12) > 50.
For the case n — 13, Theorem 2.1 gives 7^13) > 56 and Theorem 2.2 gives 7t(r13) > 59. The values of the objective function in our optimization problems having 22 variables and 8 constraints give lower bounds Y(r13) > 65 and 7t(r13) > 75.
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