CaROMaD, Volume 01 (2016) : 53 − 61
http://www.laromad.usthb.dz
Cahiers de Recherche Opérationnelle
et Mathématiques de la Décision
Identifying codes of Cartesian product of two cliques
Anissa Hissoum & Ahmed Semri
LaROMad, Fac. Maths, USTHB, PO 32, 16111 Bab Ezzouar, Algeria
[email protected], [email protected]
Abstract: An identifying code in a graph is a subset of vertices with the property that the
closed neighborhood of any vertex has a unique and nonempty intersection with the code. The
vertices of a code are called codewords.The notion of identifying code was initially introduced
by Karpovsky, Chakrabarty and Levitin in 1998 to model a fault-detection problem in multiprocessor systems.The problem to determine the smallest number of codewords in identifying
codes, if existing, in a graph is N P -hard. In this paper, we study identifying code problem in
the Cartesian product of two cliques with different sizes.
Key Words: Graph theory, Domination, Identifying code, clique, Cartesian product of two
graphes.
MSC(2010): 05C85, 68R10
Résumé : Les codes identifiants ont été définis pour la première fois par KARPOVSKY et al.
en 1998 pour modéliser un problème de détection et de localisation de processeurs défectueux
dans des réseaux multiprocesseurs. Depuis, d’autres applications ont été développées comme
dans les systèmes de localisation et de détection dans les environnements fermés munis de capteurs sans fil. Un code identifiant dans un graphe est un sous ensemble de sommets tel que deux
sommets quelconques du graphe ont leurs ensembles de voisinage fermé différents et non vides
dans le code. Trouver la cardinalité minimum d’un code identifiant dans un graphe, lorsqu’il
existe, est un problème N P -difficile. Ce papier traite le problème des codes identifiants dans
deux classes de graphes : l’hypercube et le produit cartésien de deux cliques de dimensions
différentes.
Mots clés : Graphes, Domination, Code identifiant, Graphe de Hamming, Hypercube, clique,
Produit Cartésien des graphes.
54
1
Anissa Hissoum & Ahmed Semri
Introduction
We consider a connected graph G = (V, E) with a vertex set V and an edge set E. We refer to
a distinguished subset of V as a code, and to its elements as codewords. An identifying code
in G is a nonempty subset of the vertex set V , such that all the vertices of V have different
nonempty neighborhoods in the code, ie a dominating and separating code at the same time.
The initial application for identifying codes was to fault diagnosis in multiprocessor system in
1998 by Karpovsky, Chakrabarty and Levitin [?]; and has been since fundamentally connected to
a wide range of applications, including location detection in hostile environments [?], to energy
balancing of such systems [?] and environmental monitoring [?].
A necessary and sufficient conditions for a graph G = (V, E) admits an identifying code is that
it does not contain twin vertices (two vertices sharing the same set of neighborhood). In this
case, the set of vertices V is always an identifying code of G. The problem therefore is to find
the minimum number of codewords required to identify each vertex of G. It has been shown
in [?] that this problem was N P -hard. If C is an identifying code of minimum cardinality in a
graph with n vertices, then |C| ≥ ⌈log2 (n + 1)⌉ [?, Th1] . This first lower bound comes from
the fact that with the elements of C, it is possible to build 2|C| − 1 distinct and non-empty sets,
therefore to identify the set of vertices of the graph, it is necessary that n ≤ 2|C| − 1.
We provide upper bounds on the minimum cardinality of an identifying code in the Cartesian
product of two cliques of different sizes.
2
Identifying Codes of Kn Km, m ≥ n ≥ 2
Given two graphs G and H, the Cartesian product of G and H denoted by GH, is the graph
on vertex set V (G) × V (H); and any two vertices (x1 , y1 ) and (x2 , y2 ) are adjacent in GH if
and only if either x1 x2 ∈ E(G) and y1 = y2 , or y1 y2 ∈ E(H) and x1 = x2 .
The cartesian product of two cliques Kn Km , is a matrix with n ∗ m vertices such that each
row Rx , x ∈ {1, ..., n} (resp column Cy , y ∈ {1, ..., m} is a clique Km (resp Kn ).
(1,1)
(1,m)
Km
(n,1)
(m,n)
Kn
Figure 1: Cartesian product of two cliques Kn Km
In [?, Theorem 1], it was shown that
minimum cardinality of an identifying code of the Carte the
sian product Kn Kn is equal to 3n
2 . The
proof of this theorem is based on the determination
3n
of an identifying code of cardinality 2 :
D = {(x, x)|x = 1, ...n} et A =
{(n − x + 1, x)|x = 1, ..., n−1
2 },
{(n − x + 1, x)|x = 1, ..., n2 },
si n est impair
si n est pair
55
Identifying codes of Cartesian product of two cliques
The set of vertices D ∪ A is an identifying code of Kn Kn .
(1,1)
A
(1,1)
A
D
D
Figure 2: Le code D ∪ A suivant la parité de n
We will establish upper bounds on the minimum cardinality of an identifying code in the Cartesian product of two cliques of different sizes Kn Km , m > n.
Given two cliques Kn and Km , as m > n geq2 and let a and b, respectively, the quotient and
the remainder of the Euclidean division of m by n.
2.1
Case when a ≥ 2 and 0 ≤ b ≤ n − 1
Proposition 1 Let C be a minimum identifying code of Kn Kan+b for all n, a ≥ 2 and 0 ≤
b ≤ n − 1, then :
|C| ≤ n(2a − 1) + 2b
Preuve.
The code C = A ∪ B ∪ P is an identifying code of Kn Kan+b , n, a ≥ 2 and 0 ≤ b ≤ n − 1 with
cardinality n(2a − 1) + 2b such that :
∅
if b = 0
{(i, i + an)/i ∈ {1, ..., b}} if b 6= 0
∅
B = {(i, kn − i + 1)/i ∈ {1, ..., n}, k ∈ {1, ..., a − 1}}∪
{(n − k + 1, (a − 1)n + k)/k ∈ {1, ..., b}}
 n
n
 n, 2 + jn /j ∈ {0, ..., a − 2} if 0 ≤ b ≤ 2 and n is odd
if ⌈ n2 ⌉ ≤ b ≤ n − 1 and n is odd
n, ⌈ n2 ⌉ + jn /j ∈ {0, ..., a − 1}
P =

∅
if n is even
A = {(i, i + kn)/i ∈ {1, ..., n}, k ∈ {0, ..., a − 1}} ∪
if b = 0
if b =
6 0
C is dominating set since it contains an element of each row of Kn Kan+b , b 6= 0, a ≥ 2. Now
we check that each pair of vertices (x, y), (x′ , y ′ ) ∈ Kn Kan+b is separated by C. Without loss
of generality, we suppose y ≤ y ′ .
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Anissa Hissoum & Ahmed Semri
1. If (x, y) and (x′ , y ′ ) are not in the same column or in the same row :
(a) If y ≤ n :
i. if y ′ ≤ n then
(x, (a − 1)n + x) ∈ N [(x, y)]\N [(x′ , y ′ )]
ii. if y ′ ≥ n + 1 then
(x, x) ∈ N [(x, y)]\N [(x′ , y ′ )]
(b) If y ≥ n + 1 then (x, x) ∈ N [(x, y)]\N [(x′ , y ′ )]
2. If (x, y) and (x′ , y ′ ) are in the same column :
(a) if y ≤ (a − 1)n
then (x, (a − 1)n + x) ∈ N [(x, y)]\N (x′ , y)
(b) if y ≥ (a − 1)n + 1
then (x, x) ∈ N [(x, y)]\N [(x′ , y)]
3. If (x, y) and (x′ , y ′ ) are in the same row.
Given k, k′ ∈ {0, ..., a} and j, j ′ ∈ {0, ..., n − 1} such that y = kn + j and y ′ = k′ n + j ′
(a) if y ≤ (a − 1)n
i. if x ≤ n2
A. if j = 0
then (n, y) ∈ N [(x, y)]\N [(x, y ′ )]
B. if n is odd and j = n2
then (n, y) ∈ N [(x, y)]\N [(x, y ′ )]
C. if j ≤ n2 then
(n − j + 1, y) ∈ N [(x, y)]\N [(x, y ′ )]
D. if j ≥ n2 + 1
then (j, y) ∈ N [(x, y)]\N [(x, y ′ )]
ii. if x ≥ n2 + 1
A. if j = 0
then (1, y) ∈ N [(x, y)]\N [(x, y ′ )]
B. if n is odd
and j = n2
then ( n2 , y) ∈ N [(x, y)]\N [(x, y ′ )]
C. if j ≤ n2
then (j, y) ∈ N [(x, y)]\N [(x, y ′ )]
D. if j ≥ n2 + 1 then
(n − j + 1, y) ∈ N [(x, y)]\N [(x, y ′ )]
(b) if (a − 1)n + 1 ≤ y < y ′ ≤ an.
Given j and j ′ such that y = (a − 1)n + j et y ′ = (a − 1)n + j ′
i. if j ′ ≤ x then
(j, (a − 1)n + j) ∈ N [(x, y)]\N [(x, y ′ )]
ii. if j ′ > x then
(j ′ , (a − 1)n + j ′ ) ∈ N [(x, y ′ )]\N [(x, y)]
Identifying codes of Cartesian product of two cliques
57
(c) if an + 1 ≤ y < y ′ ≤ an + b. Given j and j ′ such that y = an + j and y ′ = an + j ′
i. if j ′ ≤ x then
(j, an + j) ∈ N [(x, y)]\N [(x, y ′ )]
ii. if j ′ > x then
(j ′ , an + j ′ ) ∈ N [(x, y ′ )]\N [(x, y)]
(d) if 1 ≤ y ≤ an and an + 1 ≤ y ′ ≤ an + b. Given k ∈ {0, .., a}, j ∈ {0, .., n − 1} and
j ′ ∈ {1, .., b} such that y = kn + j et y’=an+j’
i. if x ≥ b + 1 then
(j ′ , y ′ ) ∈ N [(x, y ′ )]\N [(x, y)]
ii. if x ≤ b then
A. if x ≤ n2
• if j = 0 then
(n, y) ∈ N [(x, y)]\N [(x, y ′ )]
• if n is odd and j = n2 then
(n, y) ∈ N [(x, y)]\N [(x, y ′ )]
• if j ≤ n2 then
(n − j + 1, y) ∈ N [(x, y)]\N [(x, y ′ )]
• if j ≥ n2 + 1 then
(j, y) ∈ N [(x, y)]\N [(x, y ′ )]
B. if x ≥ n2 + 1
• if j = 0 then
(n, y) ∈ N [(x, y)]\N [(x, y ′ )]
• if n is odd and j = n2 then
( n2 , y) ∈ N [(x, y)]\N [(x, y ′ )]
• if j ≤ n2 then
(j, y ′ ) ∈ N [(x, y ′ )]\N [(x, y)]
• if j ≥ n2 + 1 then
(n − j + 1, y) ∈ N [(x, y)]\N [(x, y ′ )]
Figure 3: Identifying code of K5 K25
2.2
Case when a = 1 and b 6= 0
In this case, we propose two identifying codes according to the values of b
58
Anissa Hissoum & Ahmed Semri
Figure 4: Identifying code of K6 K24
K4 K9
K4 K10
K4 K11
Figure 5: Identifying code of Kn Kan+b , n = 4, a = 2 and b ∈ {1, 2, 3}
2.2.1
If 1 ≤ b ≤
n
2
Proposition 2 If C is a minimum identifying code of Kn Kn+b , 1 ≤ b ≤
n
2
, then :
3n
|C| ≤
+b
2
Preuve. If 1 ≤ b ≤ n2 , then the code C = A ∪ B is an identifying code of Kn Kn+b , n ≥ 2
with cardinality 3n
2 + b such that :
A = {(i, i)/i ∈ {1, ..., n}} ∪ {(i, n + i)/i ∈ {1, ..., b}}
B = {(n − i + 1, i)/i ∈ {1, ...,
n
2
}
59
Identifying codes of Cartesian product of two cliques
K5 K11
K5 K12
K5 K13
K5 K14
Figure 6: Identifying code of Kn Kan+b , n = 5, a = 2 and b ∈ {1, 2, 3, 4}
K7 K8
K7 K9
K7 K10
Figure 7: Identifying codes of Kn Kn+b , n = 7 and b ∈ {1, 2, 3}
2.2.2
If
n
2
+1≤b≤n−1
Proposition 3 If C is a minimum identifying code of Kn Kn+b with
then :
|C| ≤ n + 2b
n
2
+ 1 ≤ b ≤ n − 1,
Preuve.
If n2 + 1 ≤ b ≤ n − 1, then the code C = A ∪ B ∪ P is an identifying code of Kn Kn+b , n ≥ 2
with cardinality n + 2b such that :
60
Anissa Hissoum & Ahmed Semri
A = {(i, i)/i ∈ {1, ..., n}} ∪ {(i, n + i)/i ∈ {1, ..., b}}
B = {(n − i + 1, i)/i ∈ {1, ..., b}
P =
{ n,
∅
n 2
}
if n is odd
if n is even
K7 K11
K7 K12
K7 K13
Figure 8: Codes identifiants de Kn Kn+b , n = 7 et b ∈ {4, 5, 6}
We conjecture that the upper bounds found in the propositions ??, ?? et ?? coincide with the
exact values of the minimum cardinality of identifying codes of Kn Kan+b .
Conjecture 1 (Anissa HISSOUM, Ahmed SEMRI) Let Kn and Km ,m > n ≥ 2 be two
cliques and let a, b be the quotient and the remainder of the Euclidean division of m by n
respectively. If C is a minimum identifying code of Kn Km , then :

 n(2a − 1) + 2b
3n
|C| =
+b
 2
n + 2b
3
if a ≥ 2 and 0 ≤ b ≤ n − 1;
if a = 1 and 1 ≤ b ≤ n2 ;
if a = 1 and n2 ≤ b ≤ n − 1.
Conclusion
We have established upper bounds on the minimum cardinality of an identifying code in the
Cartesian product of two cliques with different sizes Kn Km following values of de a and b,
where m = an + b and b ≤ n − 1. All possible cases are summarized in Table ??.
Identifying codes of Cartesian product of two cliques
61
Table 1: Recapitulation
a
b
Upper Bound
a≥2
a=1
a=1
a=1
0 ≤ b ≤ n − 1
1 ≤ b ≤ n2
n
2 ≤b≤n−1
b=0
n(2a
3n − 1) + 2b
2 +b
n
+
3n 2b
(exact value[?, Theorem 1]
2
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