ARS MATHEMATICA CONTEMPORANEA
Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 13 (2017) 367-377
The binary locating-dominating number of some
convex polytopes
Ana Simic
Faculty of Mathematics, University of Belgrade, Študentski trg 16, 11000 Belgrade, Serbia
Milena Bogdanovic
Pedagogical Faculty in Vranje, University ofNis, Partizanska 14, 17500 Vranje, Serbia
Jelisavka Milosevic
FASPER, Visokog Stevana 2, 11000 Belgrade, Serbia
Received 11 November 2015, accepted 14 March 2017, published online 23 March 2017
In this paper the binary locating-dominating number of convex polytopes is considered. The exact value is determined and proved for convex polytopes Dn and Rn, while for the convex polytopes Rn, Qn and Un a tight upper bound of the locating-dominating number is presented.
Keywords: Locating-dominating number, convex polytopes. Math. Subj. Class.: 05C69, 05C90
1 Introduction
Let G be a simple connected undirected graph G = (V, E), where V is a set of vertices, and E is a set of edges. The open neighborhood of a vertex v e V is NG(v) = {u e V | (u, v) e E} and the closed neighborhood is NG[v] = {u e V | (u, v) e E} U {v}. We write N(v) or N[v] if the graph G is clear from the context [4]. For a graph G = (V, E) a dominating set is a vertex set D C V such that the union of the closed neighborhoods of the vertices in D is all of V; that is, N[D] — V. Equivalently, each vertex not in D is adjacent to at least one vertex in D, e.g. for every vertex v e V \ D, N(v) n D = 0. The
E-mail addresses: simicana4as@gmail.com (Ana Simic), mb2001969@beotel.com, milenab@ucfak.ni.ac.rs (Milena Bogdanovic), jelisavkam@gmail.com (Jelisavka Milosevic)
Abstract
©® This work is licensed under http://creativecommons.org/licenses/by/3.0/
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domination number of G, denoted by y(G), is the minimum cardinality of a dominating set of G.
The concept of a dominating set can also be studied through assigning a weight of 1 will be assigned to the vertices in D and a weight of 0 to the vertices of V\D. In this case, D is a dominating set of G if for every vertex in G the sum of weights for closed neighborhoods is at least 1, i.e. | N [v] n S | > 1 for each v e V .A dominating set S C V is a binary locating-dominating set if for every two different vertices u,v e V \ S holds N(u) n S = N(v) n S ([12]). The binary locating-dominating number ofG, denoted by 7l-d(G), is the minimum cardinality of a binary locating-dominating set. In the sequel all terms about the locating-dominating number or set is denoted by binary locating-dominating number or set.
The article [11] studies the smallest cardinalities of locating-dominating codes on chains and cycles and the extreme values of the cardinality of a minimum r-identifying or r-locating-dominating code in any connected undirected graph G having a given number, n, of vertices is studied in [8]. For more information about these issues, see [3, 9, 14, 15, 21]. The authors of the papers [23, 24, 27] study the single-fault-tolerant locating-dominating sets and an open neighborhood locating-dominating sets in trees. More information on locating-dominating sets can be found in [12, 13, 15, 25].
The identifying code problem and binary locating-dominating problem are NP-hard in a general case [6,7]: I. Charon et al. proved in [7] that, given a graph G and an integer k, the decision problem of the existence of an r-identifying code, or of an r-locating-dominating code, of size at most k in G, is NP-complete for any r.
The comprehensive list of papers related to identifying code and binary locating-domi-nating problems were given in [19].
The following theorem gives a tight lower bound of binary locating-dominating number on regular graphs:
Theorem 1.1 (Slater [26]). IfG is a regular graph of degree r, then
'2 •IV(G)| -
Yi-d(G) >
r + 3
Graphs of convex polytopes were introduced by Baca [1]. The classes of convex poly-topes Qn and Rn were introduced in [2]. The metric dimension of convex polytopes Dn, Qn and Rn are equal to 3, as was proved in [16]. In [17] it was proven that metric dimension of convex polytopes Sn, Tn and Un is also equal to 3. Minimal doubly resolving sets and the strong metric dimension of convex polytopes Dn and Tn are studied in [18]. M. Salman et al. [22] were considering three similar optimization problems: the fault-tolerant metric dimension problem, the local metric dimension problem and the strong metric dimension problem of two convex polytopes Sn and Un.
2 A modified integer linear programming formulation
An integer linear programming (ILP) formulation of minimum identifying code problem was given in [5]. If S is an identifying set, then decision variables xj are defined as:
(1, i e S
Xj [0, i e s
Then, the ILP formulation of minimum identifying code problem from [5] is presented as follows:
Ana Simic et al.: The binary locating-dominating number of some convex polytopes 369
min ^^ Xi
iev
subject to
E xj > i,
jew [i]
E Xj >l,
jew [i] vw [k]
xi e {0, i},
i e V
i, k e V, i = k i € V
(2.1)
(2.2)
(2.3)
(2.4)
The objective function (2.1) ensures that the identifying code set has a minimal cardinality, and constraints (2.2) defines S to be a dominating set. Identifying feature is represented by constraints (2.3) while the binary nature of decision variables xi are given by constraints (2.4).
This formulation can not be directly used for the binary locating-dominating problem. Therefore, it needs to be adapted by changing constraints (2.3) into the constraints (2.5).
Xi + Xk + E
jew (i)VN (k)
xj >1
i, k e V, i = k
(2.5)
Constraints (2.3) and (2.5) are the same when vertices i and k are not neighbors, e.g. N[i]VN[k] = {i, j} U (N(i)VN(k)). The change between (2.3) and (2.5) is reflected only when vertices i and k are neighbors, i.e. i e N(k). Then, by constraints (2.5), at least one of vertices i, k or some j e N(i) VN(k) must be in S. When i and k are not neighbors, then N[i]VN[k] = {i, j} U (N(i)VN(k)), so constraints (2.3) and (2.5) are equal.
In [28] it was noted that if d(u, v) > 3 then u, v has no neighbors in common, therefore, N(u) n S = N(v) n S need not be checked for equivalence. This becomes computationally important for large graphs as it allows us to minimize the number of constraints generated by the locating requirement. Using this idea, constraints (2.5) would be further improved:
Xi + Xk +	Xj > 1, i, k e V, i = k, d(i, k) < 2	(2.6)
jew(i)vw(k)
The proposed formulation with a reduced number of constraints can be used to find the exact optimal values for problems of small dimensions. Moreover, as it can be seen from [10], ILP formulation can be tackled by efficient metaheuristic approaches for obtaining suboptimal solutions for large dimensions.
3 The exact values
3.1 Convex polytope Dn
The graph of convex polytope Dn, on Figure 1, was introduced in [16]. It consists of 2n 5-sided faces and a pair of n-sided faces. Mathematically, it has vertex set V(Dn) = {«i, bi, ci,di | i = 0,1,... ,n - 1} and edge set E (D„) = {(^,^+1), (di, di+i), (ai, bi), (bi, ci), (ci, di), (bi+1, ci) | i = 0,1,..., n - 1}. Note that arithmetic in the subscripts is performed modulo n.
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Figure 1: The graph of convex polytope Dn
Table 1: Locating-dominating vertices in Dn.
d
-3
n	v € V \ S	Sf)N [v]	v € V \ S	SRN [v]
3 k	asi	{asi+i, bsi}	asi+2	{asi+i}
	bsi+i	{asi+i, csi+i}	bsi+2	{csi+i}
	Csi	{bsi}	Csi+2	{bS(i+1), dsi+2}
	d3i	{ds(i-1)+2}	dsi+i	{cSi+1, dsi+2}
3k + 1	asi	{aS(i-1) + 2, bsi}	asi+i	{asi+2}
	bsi+i	{csi+i}	bsi+2	{asi+2, csi+i}
	csi	{bsi, dsi}	csi+2	{bs(i+i)}
	dsi+1	{csi+l, dsi}	dsi+2	{ds(i+i)}
	ask	{bsk, as(k-1)+2}	csk	{bsk, dsk}
	ao	{bo}		
3k + 2	a3i+1	{asi, bsi+i}	asi+2	{as(i+i)}
	bsi	{aSi, cs(i-1) + 2}	bsi+2	{csi+2}
	csi	{bsi+i, dsi}	csi+i	{bsi+i}
	dsi+i	{dsi}	dsi+2	{cSi+2, ds(i+1) }
	bsk	{ask, cs(k-1)+2}	Csk	{bsk+i, dsk}
	ask+i	{ask, bsk+i}	csk+i	{bsk+i}
	dsk+i	{dsk }	bo	{ao}
Ana Simic et al.: The binary locating-dominating number of some convex polytopes 371
Theorem 3.1.
Yi-d(D„) =
4 • n
Proof. Firstly, notice that Dn is a regular graph of degree 3, with 4n vertices. Then, by
Theorem 1.1 it holds Yl-d(Dn) > Let
2-4-n
3+3
m.
{a3i+i,&3i,C3i+i, d3i+2|i = 0,. .. ,k - 1}, S = ^ {&3fc, d3fc} (J{a3i+2, &3i, C3i+1, d3i|i = 0,. .
,k - 1},
n = 3k n = 3k + 1
„{a3fc,&3fc+i,d3fc} U{«3i, b3i+i,C3i+2,d3i|i = 0,. .., k - 1}, n = 3k + 2
Now, let us prove that S is a locating-dominating set of Dn. In order to do that, we
need to consider three possible cases:
Case 1: n = 3k. As can be seen from Table 1, neighborhoods of all vertices in V \ S and their intersections with set S are non-empty and distinct. Although some formulas for some intersections can be somewhat similar, they are distinct. For example, Sf|N[0^+2] = {a<3(i+i)} = {a3i+i} = Sf|N[634+1], since indices 3(i + 1) = 3i + 3 = 3i + 1. Similarly, Sf|N[d3i+i] = ^¿+2,4»} = W+2, d3(»+i)} =
SDN [d3i+2];
Case 2: n = 3k +1. As in the previous case, once again, all intersections of neighborhoods N[v] with set S, i.e. S p| N[v], are non-empty and distinct. This also can be seen from Table 1;
Case 3: n = 3k + 2. As in both presvious cases, once again, all intersections of neighborhoods N[v] with set S, i.e. Sp| N[v], are non-empty and distinct, which also can be seen from Table 1.	□
3.2 Convex polytope R^
The graph of convex polytope on Figure 2 is introduced in [20]. It has vertex set
V	= {Oi, 6i,Ci, di, e», fi | i = 0,... ,n -1} and edge set E = {(o^oi+i), (0» ,6»), (&i,q), (6i+i, ci), (ci, di), (di, ei), (d»+i, e»), (e», /¿), (/¿, /¿+i) | i = 0,. .., n - 1}.
Theorem 3.2.
7,_d(Rn) = 2 • n.
Proof. It can be seen that R^ is a regular graph of degree 3, with 6n vertices. Then, by Theorem 1.1 it holds Yl-d(Rn) > -|+:3r =2 • n. Now, let us prove that a set S = {6i,ei | i = 0,..., n - 1} is a binary locating-dominating set of Rn. Indeed, it is easy to see that all intersections Sf|NM = M; Sf|N[c»] = {6i,6i+i}; Sf|N[di] = {ei-i,ei} and S Pi N[/] = {e»} are non-empty and distinct. Since S is a binary locating-dominating set of Rn and |S| =2 • n therefore, Yl-d(Rn) < 2 • n. Due to the previously proved fact that 7l-d(Rn) > 2 • n, it is proven that Yl-d(Rn) is equal to 2 • n.	□
4 The upper bounds
4.1 Convex polytope Qn
The graph of convex polytope Qn in Figure 3, is introduced in [2]. It has vertex set
V	(Qn) = H,&i,ci,di | i = 0,1,.. .,n-1} and edge set E (Qn) = {(«i ,«i+i), (&iA+i),
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Ars Math. Contemp. 13 (2017) 275-291
e1 d0
f 0
f 2M
fn-2
Figure 2: The graph of convex polytope R^
(di ,di+i), (ai,bi), (bi,Ci), (cj,dj), (&i+i,cj) | i = 0, 1,...,n - 1}. We call the cycle induced by set of vertices {a0, a1,..., an-1} the inner cycle, the cycle induced by {do, d1,..., dn-1} the outer cycle, and the middle cycle are induced by set of vertices {b0, b1,..., bn-1}. This polytope consists of n 5-sided faces, n 4-sided faces and n triangles.
Figure 3: The graph of convex polytope Qn.
Theorem 4.1.
Yl-d(Qn) <
4 • n
and this bound is tight.
d
b
0
d
2
d
d
2

Ana Simic et al.: The binary locating-dominating number of some convex polytopes 373 Table 2: Additional data for Qn compared to Dn.
n	v G V \ S	Sf! N[v]	v G V \ S	sn n [v]
3k	b3j+i	{a3j+i, b3j, C3j+i}	b3j+2	{b3(j+1h c3j+1}
3k + 1	b3j+i	{b3j, C3j+i}	b3j+2	{a3j+2, b3(j+1^ c3j+1}
3k + 2	b3j	{a3j, b3j+^ c3(j-1)+2}	b3j+2	{b3j+1, C3j+2}
	b3fc	{a3k, c3(k-1)+2}	bo	{ao, bi, b3k+i}
Proof. Let
f{a3i+i,&3i,C3i+i, d3i+2|i = 0,. .. ,k - 1},	n = 3k
{b3k, 4k} U{a3i+2,b3j, C3i+i, d3i|i = 0,. .., k - 1},	n = 3k + 1 {a3k,&3fc+i,d3fc} U{a3i, b3i+i,C3i+2,d3i|i = 0,. .., k - 1}, n = 3k + 2
Note that this set is the same as for convex polytopes Dn. This is not a surprise, since convex polytopes Qn have only n additional edges (bj, bj+i), i = 0,..., n - 1 compared to Dn. Therefore, except vertices bj, i = 0,..., n -1, all neighborhoods of vertices in V\S and their intersection with set S are the same as in Table 1. Additional data is presented in Table 2.
As can be seen from Table 3 and additional data from Table 2, in all three cases, neighborhoods of all vertices in V \ S and their intersection with set S are non-empty and distinct. Therefore set S is a locating-dominating set for Qn. Since |S| = therefore,
Yl-d(Qn) < \1.
Using the CPLEX solver on the integer linear programming formulation (2.1), (2.2), (2.4), and (2.6) we have obtained optimal solutions: 7;-d(Qs) = 7, 7;-d(Q6) = 8, 7i-d(Qr) = 10, ..., 7i-d(Q2s) = 38, 7i-d(Q29) = 39 and 7i-d(Q3o) = 40 which all match the proposed upper bound in this theorem. Therefore, the proposed upper bound is tight.	□
4.2 Convex polytope Rn
The graph of convex polytope R„, on Figure 4, has been introduced in [2]. It has vertex
set V = {aj, b, cj | i = 0,..., n - 1} and edge set E = {(a.j, aj+1), (a,j, b), (aj+1, bj), (b, bj+1), (bj, cj), (cj, cj+1) | i = 0,..., n - 1}. This graph consists of n 4-sided faces and 2n triangles.
Theorem 4.2.
7i-d(Rn) < n,
and this bound is tight.
Proof. Let S = {bj | i = 0,..., n - 1}. It is easy to see that all intersections S p| N[a] = {bj-i, bj} and S p| N[cj] = {bj} are non-empty and distinct. Since S is a binary locating-dominating set of Rn and |S| = n therefore, 7;-d(Rn) < n.
Using the CPLEX solver on integer linear programming formulation (2.1), (2.2), (2.4), and (2.6), we have obtained optimal solutions. For 5 < n < 31, 7;-d(Rn) = n, which match the proposed upper bound in this theorem. Therefore, the proposed upper bound is tight. □
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Figure 4: The graph of convex polytope E„.
2
-3
4.3 Convex polytope Un
Mathematically, the graph of convex polytope Un, on Figure 5, introduced in [17], has vertex set V = {ai, hi, ci,di,ei | i = 0, ...,n - 1} and edge set E = {(cm, ai+i), (ai, hi), (bi,bi+1), (hi, ci), (ci,di), (ci+i, di), (di, eH), (ei,e+) | i = 0,...,n - 1}. This graph in Figure 5 has 2n 5-sided faces and n 4-sided faces.
Theorem 4.3.
and this bound is tight.
Yl-d(Un) <
5 • n
Proof. If n = 3k, let S = {a3i+1,h3i, c3i+1, d3i+2,e3i | i = 0,...,k - 1}, if n = 3k + 1 let S = {a3k, c3k, e3k} U{a3i, h3i+2, c3i, d3i+i, e3i | i = 0,... ,k - 1} and if n = 3k + 2 let S = {a3k+i, h3k, c3k+i, e3k+i} U{a3i+i Mi, c3i+i, d3i+2, e3+ | i = 0,...,k - 1}.
Now, let us prove that S is a locating-dominating set of Un. In order to do that, as we did in proofs of previous Theorems, we need to consider three possible cases. As it can be seen from Table 3, in all three cases, neighborhoods of all vertices in V \ S and their intersection with set S are non-empty and distinct.
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Using the CPLEX solver on the integer linear programming formulation (2.1), (2.2), (2.4), and (2.6) we have obtained optimal solutions: 7l-d(U5) = 9, 7l-d(U6) = 10, 7i-d(Ur) = 12, ..., 7i-d(U22) = 38, Yi-d^) = 39 and 7,^(^24) = 40 which all match the proposed upper bound in this theorem. Therefore, the proposed upper bound is tight. □
Table 3: Locating-dominating vertices in Un.
n	v € V \ S	Sf)N [v]	v € V \ S	SDN [v]
3k	a3i	{a3i+i, b3i}	a3i+2	{a3i+1) }
	b3i+i	{a3i+i, b3i, C3i+i}	b3i+2	{b3(i+1)}
	C3i	{b3i, d3(i-1)+2}	c3i+2	{d3i+2}
	d3i	{C3i+i, e3i}	d3i+1	{c3i+1}
	e3i+i	{e3i}	e3i+2	{d3i+2, e3(i+1)}
3k + 1	«3i+i	{a3i}	a3i+2	{a3(i+1), b3i+2}
	b3i	{a3i, b3(i-1)+2, c3i}	b3i+1	{b3i+2}
	C3i+1	{d3i+i}	C3i+2	{b3i+2, d3i+i}
	d3i	{c3i, e3i}	d3i+2	{c3(i+1)}
	e3i+i	{d3i+i, e3i}	e3i+2	{e3(i+1)}
	&3fc	b3(fc-1) + 2, c3k}	d3k	{C3k, e3k}
	bo	{ao, co}		
3k + 2	a3i	{a3i+i, b3i}	a3i+2	{a3i+i}
	b3i+i	{a3i+i, b3i, C3i+i}	b3i+2	{b3(i+1)}
	C3i	{b3^ d3(i-1)+2}	C3i+2	{d3i+2}
	d3i	{c3i+i}	d3i+1	{c3i+1, e3i+1}
	e3i	{e3i+i}	e3i+2	{d3i+2, e3i+i}
	«3fc	{a3k+i, b3k}	C3k	{b3k, d3(k-1)+2}
	d3k	{C3k+i}	e3k	{e3k+i}
	b3k+i	{a3k+i, bo, b3k, C3k+i}	d3k+i	{c3k+i, e3k+i}
	ao	^^ bo}	co	{bo}
	eo	{ei, e3fc+i}		
5 Conclusions
In this paper, we are studying the locating-dominating sets and the binary locating-dominating number of some convex polytopes. We are dealing with some classes of convex polytopes by considering classes: Dn, Rn, Rn, Qn and Un. For Dn and Rn exact values are obtained and proved, while for Rn, Qn and Un tight upper bounds are given.
Future work can be directed towards determining a binary locating-dominating set of some other challenging classes of graphs. The other promising direction for future work is solving of some other similar graph problem on convex polytopes.
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