The fault-tolerant metric dimension of the king’s graph Текст научной статьи по специальности «Математика»

UDC 519.8

Вестник СПбГУ. Прикладная математика... 2017. Т. 13. Вып. 3

R. V. Voronov

THE FAULT-TOLERANT METRIC DIMENSION OF THE KING'S GRAPH

Petrozavodsk State University, 33, Lenina pr., Petrozavodsk,

185910, Russian Federation

The concept of resolving the set within a graph is related to the optimal placement problem of access points in an indoor positioning system. A vertex w of the undirected connected graph G resolves the vertices u and v of G if the distance between vertices w and u differs from the distance between vertices w and v. A subset W of vertices of G is called a resolving set, if every two distinct vertices of G are resolved by some vertex of w £ W. The metric dimension of G is a minimum cardinality of its resolving set.The set of access points of the indoor positioning system corresponds to the resolving set of vertices in the graph.The minimum number of access points required to locate each of the vertices corresponds to the metric dimension of graph. A resolving set W of the graph G is fault-tolerant if Wminus{w} is also a resolving set of G, for each w £ W. The fault-tolerant metric dimension of the graph G is a minimum cardinality of the fault-tolerant resolving set. In the indoor positioning system the fault-tolerant resolving set provides correct information even when one of the access points is not working. The article describes a special case of a graph called the king's graph, or the strong product of two paths.The king's graph is a building model in some indoor positioning systems. In this article we give an upper bound for the fault-tolerant metric of the king's graph and a formula for a particular case of the king's graph. Refs 20. Figs 2.

Keywords: fault-tolerant metric dimension, strong product graphs, king's graph, access points of indoor positioning system.

Р. В. Воронов

ОТКАЗОУСТОЙЧИВАЯ МЕТРИЧЕСКАЯ РАЗМЕРНОСТЬ ГРАФА ХОДОВ ШАХМАТНОГО КОРОЛЯ

Петрозаводский государственный университет, Российская Федерация, 185910, Петрозаводск, пр. Ленина, 33

В некотором приближении аналогом задачи оптимального размещения точек доступа системы внутреннего позиционирования служит задача определения метрической размерности графа и построения его разрешающего множества. Пусть вершина w неориентированного связного графа G различает вершины u и v графа G, если расстояние между вершинами w и u отличается от расстояния между вершинами w и v. Подмножество W вершин графа G называется разрешающим, если для каждой пары вершин u и v графа G найдется различающая их вершина w £ W. Метрическая размерность графа — это минимальное число вершин в разрешающем подмножестве. Точкам доступа системы внутреннего позиционирования соответствует разрешающее множество вершин графа, а минимально необходимому числу точек доступа — метрическая размерность графа. Разрешающее множество называется отказоустойчивым, если оно остается разрешающим, даже если из него удалить любую его вершину. Отказоустойчивая метрическая размерность графа — это минимальное число вершин в отказоустойчивом разрешающем подмножестве, что в системе внутреннего позиционирования соответствует возможности определения местоположения объекта даже в случае потери информации от одной из точек доступа. Рассмотрен один частный случай графа — сильное произведение двух простых цепей, называемое иначе графом ходов шахматного короля. Установлена верхняя граница для отказоустойчивой

Voronov Roman Vlo,d,imirovich — PhD of technical sciences, associate professor;

rvoronov76@gmail.com

Воронов Роман Владимирович — кандидат технических наук, доцент; rvoronov76@gmail.com © Санкт-Петербургский государственный университет, 2017

метрической размерности графа ходов короля и приведена формула для одного частного случая. Библиогр. 20 назв. Ил. 2.

Ключевые слова: отказоустойчивая метрическая размерность, сильное произведение графов, граф ходов короля, точки доступа системы внутреннего позиционирования.

Introduction. The concepts of the graph theory is used to describe the problem of navigation in the network [1] and in indoor positioning system to model the floorplan of the building. The building floorplan is modeled by the undirected connected graph G = (V, E), where the vertices of the set V represent small zones, and the edges of the set E denote the possibility of moving directly between zones. A zone may consist of only one room, and big rooms may be partitioned into several zones. The distance d(u,v) between vertices u and v is the minimum number of edges in the path having these two vertices as its endpoints.

In some vertices of the graph we can place the landmarks of the navigation system or access points of the indoor positioning system [2]. The set of access points of the indoor positioning system corresponds to the resolving set of vertices in graph. The minimum number of access points required to locate each of the vertices is called the metric dimension.

Formally, let W = {wi, ...,wk} be an ordered subset of vertices of graph G. The ordered fc-tuple r(v | W) = (d(v, wi),..., d(v, wk)) is called a representation of the vertex v with respect to W. The subset of vertices W с V is called a resolving set, if every two vertices u, v have distinct representations r(u | W) and r(v | W). The metric dimension (3(G) of the graph G is a minimum cardinality of the resolving set for G. A resolving set with the minimum number of vertices is called a metric basis for G.

In other words, the metric dimension of the graph G is the smallest integer m, for which subset W с V exists, such that IW| = m and for every pair of vertices u,v e V there is w e W, that the distance between the vertices w and u is not equal to the distance between the vertices w and v. We also will say, that a vertex w of the graph G resolves the vertices vi and v2 in G (is able to distinguish vi and v2), if d(w, vi) = d(w, v2).

A resolving set W of the graph G is fault-tolerant if W \ {w} is also a resolving set of G, for each w e W. The fault-tolerant metric dimension (3'(G) of G is a minimum cardinality of the fault-tolerant resolving set. A fault-tolerant resolving set of cardinality (3'(G) is called a fault-tolerant metric basis of G.

The strong product Gi H G2 of the graphs Gi = (Vi, Ei) and G2 = (V2, E2) is the graph G = (V,E), such that V = Vi x V2 and two distinct vertices (ui,u2) and (vi,v2) are adjacent in G if and only if ui = vi and (u2,v2) e E2, or u2 = v2 and (ui,vi) e Ei, or (ui,vi) e Ei and (u2,v2) e E2.

Now we denote Pm = (Im,Jm) — path graph, where m is natural number, Im = {l,...,m} and Jm = {(i, i +1) | i = 1,...,m — 1}.

The king's graph with natural parameters (m,n) is a graph Pm H Pn, that represents all legal moves of the king chess piece on a m x n chessboard. The vertex set of the m x n king's graph is the Cartesian product V = Im x In. It is easy to check, that d(vi7 v2) = max{|ii — i2 ^ Hi — j21} for any two vertices vi = (ii7 ji) and v2 = (i2, j2) of graph Pm H Pn.

It is known that e(Pn H Pn) = 3 [3], fj'(Pn H Pn) = 4 [4] for n > 2. The following theorem is proved in [5].

Theorem 1 [5]. For any integers n and m such that 2 ^ m < n,

l3(Pm H Pn ) =

n + m — 2 m — 1

Figure 1 shows graph P3 K P12. Vertices of metric basis are black, /3(P3 K P12) = 7.

Fig. 1. Metric basis for graph P3 K P12

In this paper we study the problem of finding a sharp bound for the fault-tolerant metric dimension of the king's graph and the exact value for a particular case. We assume that n > m for the graph Pm K Pn. The case m > n is considered analogously.

Related works. The problems of finding the metric dimension of a graph were introduced independently by Slater (1975) and Harary and Melter (1976) [6, 7]. Melter studied the metric dimension problem for the tree. Garey and Johnson (1979) noted that determining the metric dimension of the graph is an NP-complete problem. Khuller, Raghavachari, Rosenfeld (1996) described the application of the metric dimension problem in the field of computer science and robotics and outlined the graphs with metric dimension 1 and 2 [8]. Chartrand, Eroh, Johnson, Oellermann (2000) described the application in chemistry [9]. The strong metric dimension problem was introduced by Sebo and Tannier (2004) [10]. Fehr, Gosselin, Oellermann (2006) studied the metric dimension for different types of graphs, for exsample Cayley digraphs [11]. The concept of the fault-tolerant metric dimension was introduced by Hernando, Mora, Slater, Wood (2008) [12]. Okamoto, Phinezy, Zhang (2010) introduced the concept of local metric dimension [13]. The metric dimension of the random graph was considered by Bollobas, Mitsche, Pralat (2012) [14]. The formulas for metric dimension of many graph classes were studied [15-17]. Zejnilovic, Mitsche, Gomes and Sinopoli (2016) extended the metric dimension to the graphs with missing edges [18].

The main results. We present the main result in the form of two theorems.

The first theorem gives the upper bounds for the fault-tolerant metric dimension of the king's graph.

Theorem 2. For any integers n and m, such that 2 ^ m < n, the following assertion hold. If (m — 1) is a divisor of (n — 2), then

otherwise

j3'(Pm H Pn) < 2

в' (Pm H Pn) < 2

n - 2 m1

n — 1 m1

2.

There is a formula for the fault-tolerant metric dimension for a partcular case of king's graph in second theorem.

Theorem 3. For any integers n and m, such that m is even, m ^ 2, n ^ 2m — 1 and (m — 1) is a divisor of (n — 1),

e'(Pm

IPn) = 2

n — 1 m1

2.

First we introduce some definitions and prove some lemmas.

For the integers m, n, j, such that 2 < m < n and j G {1,... ,n}, Vj denotes the vertex subset of graph Pm K Pn, where Vj = {(i,j) | i = 1,..., m}. We introduce the notation

Vj! j = U Vj.

Lemma 1. Let 2 ^ m < n be integers. Let W be a resolving set of graph G = Pm KPn. If vi = (ii, j'), v2 = (i2, j') are vertices of G and w = (i, j) G W, such that lj — j 'I ^ m — 1, then vertex w does not resolve the vertices v1 and v2.

Proof. Since li — i'l ^ m — 1 for all i' = 1,...,m, then

d(w, vi) = d(w, v2) = lj — j 'I.

Hence, then vertex w does not resolve the vertices v1 and v2. Lemma is proved.

Lemma 2. Let 2 ^ m < n be integers. Let W be a resolving set of graph Pm K Pn. Then for any j' G{1,...,n} there exists vertex w = (i, j) G W, such that lj — j'l < m — 1.

Proof. Suppose, for the contrary, that exists j' G {1,... ,n}, such that for all w = (i, j) G W we have lj — j'l > m — 1. We now take any distinct i1, i2 G {1,...,m}. According to the Lemma 1, no vertex w G W resolve the vertices v1 = (i1, j') and v2 = (i2, j'). This contradiction proves the lemma.

Lemma 3. Let 2 ^ m < n be integers and let W be a resolving set of graph Pm K Pn. Let j' g{1, ..., n}. If there exists only one vertex w = (i, j) G W, such that lj —j'l < m—1, then j = j'.

Proof. Suppose, for the contrary, that exists j' g{1,... ,n}, that there exists only one vertex w = (i, j) G W, such that lj — j'l < m — 1, but j = j'.

Let i1 = i. If i < m, then let i2 = i + 1. If i = m, then let i2 = i — 1. Let v1 = (i1, j'), v2 = (i2, j'). Then d(w, vi) = lj — j'l, d(w, v2) = lj — j'l, hence d(w, vi) = d(w, v2) and vertex w G W does not resolve the vertices v1 and v2. In addition, according to the Lemma 1, no vertex in W \ {w} that distinguish vertices v1 and v2.

This contradiction proves the lemma.

Lemmas 2 and 3 lead to the next results.

Corollary 1. Let 2 ^ m < n be integers and let W be a resolving set of graph Pm KPn. Then for all j G {1,...,n} there exists w G W, such that w G Vj or exist two distinct vertices (i1, ji) G W and (i2, j2) G W, that lj — j1l < m — 1 and lj — j2l <m — 1.

Corollary 2. Let 2 ^ m < n be integers and let W be a fault-tolerant resolving set of graph Pm K Pn. Then for all j G {1,...,n} there exist two distinct vertices wi,w2 G W, such that wi,w2 G Vj or exist three distinct vertices (ii, ) G W, (i2, j2) G W and (i3, j3) G W, that lj — jil < m — 1, lj — j2l <m — 1 and lj — j31 <m — 1.

Lemma 4 [19]. A resolving set W of a graph G is fault-tolerant if and only if every pair of vertices in G is resolved by at least two elements of W.

Lemma 5. Let G = Pm K Pn, where m is even, n ^ m ^ 2. Let W be a fault-tolerant resolving set of G. Then for all j G {0,...,n — m + 1},

Vj+i,j+m-if) W

> 2.

Proof. Let j G {0,...,??— m + 1}, 1/' = Vj+1, j+m-U = (f,J + f), v2 + 1 + In this case v2 G V'. Let V be the vertex set of graph G.

j=ji

By Lemma 4 for the vertices vi, v2 there exist wi,w2 G W, wi = w2, such that d(wi,vi) = d(w\,v2) and d(w2,v\) = d(w2,v2).

Consider a vertex w = (i,k) G V \ V. Since \i — Щ-l <

-11 ^

and

r 2 I 2

|k - j - YI > f- we have = |/c - j - and d.(w,v2) = |/c - j - Hence

d(w, vi) = d(w, v2) and no vertex in V \ V' is able to distinguish vi and v2.

Thus wi,w2 G V'p| W. Therefore, the proof is complete.

Lemma 6. Let G = Pm K Pn, where m ^ 2 and n ^ 2. Let W be a fault-tolerant resolving set of G and let j G {1,...,n}. If exist distinct vertices wi,w2 G W, that W = {wi,w2} or d(w,vi) = d(w,v2) for any w G W \ {wl:w2} and for each pair of distinct vertices v\,v2 G Vj, then wl,w2 G Vj.

Proof. Let G, j, w\ and w2 be as in the hypotheses. Let V be the vertex set of graph G. By Lemma 4 for every pair of vertices vl ,v2 G Vj there are at least two vertices of W, which are able to distinguish vl and v2. Hence, we have that d(wl,vl) = d(wl,v2) and d,(w2,v\) = d(w2,v2) for all different v\,v2 G V.

We will show that wl G Vj. Suppose, for the contrary, that wl = (i,ji) and j\ = j. If i = 1, then w\ is not able to distinguish vi = (1,j) G Vj and v2 = (2, j) G Vj. If i > 1, then w\ is not able to distinguish vi = (i,j) G Vj and v2 = (i — 1,j) G Vj. In both cases we have d,(w\,vi) = d(wl,v2) = \j\ — j\ and we get a contradiction.

The proof that w2 G Vj is deduced analogously.

Lemma is proved.

Lemma 7. Let G = Pm KPn, where m ^ 2 and n ^ 2m — 3. Let W be a fault-tolerant resolving set of G and let Vj = Vmax{i,j-m+2}, min{n,j+m-2}, where j G{1,...,n}. If

VR W

Vjf) W. Let V be the vertex set of

then Vjp| W c Vj.

Proof. Let j G {1,...,n} and let {w\, w2} graph G. We differentiate two cases for V \ Vj.

Case 1: V \ Vj = 0. Consider a vertex w = (i,k) G V \ Vj and any different vertices vi = (il, j) G Vj, v2 = (i2,j) G Vj. Since \i — i\\ ^ m — 1, \i — i2\ ^ m — 1 and \k — j\ ^ m — 1 we have d(w,vi) = \k — j\ and d(w,v2) = \k — j\. Hence d(w,vi) = d(w,v2) and no vertex in V \ Vj is able to distinguish vi and v2. Thus no vertex in W \ {wi,w2} is able to distinguish any two different vertices u,v G Vj. Therefore, by Lemma 6 wi ,w2 G Vj.

Case 2: V \ Vj = 0. In this case we have W = {wi,w2}, by Lemma 6 wi,w2 G Vj and we conclude the proof.

Lemma 8. Let G = Pm K Pn, where m ^ 2 and n ^ m. Let W be a fault-tolerant resolving set of G. Then

Vn

n-m+1, n

W

> 3.

Proof. By Corollary 2 we have \Kp| W \ > 2 or \Vn-m+2,n{\ W \ > 3 and, additionally, \Vn-if] W\ > 2 or \Vn-m+i, ^ W\ > 3. Anyway we get \Vr-m+i, ^ W\ > 3. The lemma is proved.

Now we present the proof of the Theorem 2.

Proof. Let 2 < m < n be integers and let G = Pm K Pn. In paper [20] is shown, how construct a resolving set W (a metric generator) for graph G, such that

\W | = к =

n — 1 m1

+ 1.

We use that construction and consider two cases.

Case 1: (m — 1) is a divisor of (n — 2). Let

wit

w2t =

(1, min{n, (t — 1)(m — 1) + 1}), (m, min{n, (t — 1)(m — 1) + 1}),

t =1,...,k,

(m, min{n, (t — 1)(m — 1) + 1}), (1, min{n, (t — 1)(m — 1) + 1}),

t = 1,...,k- 1,

if t is odd, otherwise,

if t is odd, otherwise,

w2k

(1, n), if к is odd, (m, n), otherwise,

w3t

(1, min{n, (t — 1)(m — 1) + 1}), (m, min{n, (t — 1)(m — 1) + 1}),

t = 1,...,k- 1,

if t is odd, otherwise,

w3k =

(1, n — 1), if k is odd, (m, n — 1), otherwise.

For i = 1,

,3 let

Wi = {wit | t =1,...,k} .

Wi is resolving sets of G [20]. Analogously we can show that W2, W3 are resolving sets of G.

Let U = {wik-i,wik,w3k}. It is obviously, that wik-i = w3k-i, wik = w2k, w3k = w2k - i, Wi \ U = W3 \U, (Wi \ U )n(W2 \U) = 0. Let W = Wi U W2 U W3 .We differentiate five cases. If w G Wi \ U, then W2 c W \ {w}. If w G W2 \ U, then Wi c W \ {w}. If w = wik-i, then W2 c W \ {w}. If w = wik, then W3 c W \ {w}. If w = w3k, then Wi c W \ {w}. We can point out that, in any case, W \ {w} is resolving sets of G. Thus W is a fault-tolerant resolving set.

Case 2: (m — 1) is not a divisor of (n — 2). Let

wit

w2t

For i = 1, 2 let

(1, min{n, (t — 1)(m — 1) + 1}), (m, min{n, (t — 1)(m — 1) + 1}),

(m, min{n, (t — 1)(m — 1) + 1}), (1, min{n, (t — 1)(m — 1) + 1}),

t = 1,...,k.

Wi = {wit | t =1,...,k} .

if t is odd, otherwise,

if t is odd, otherwise,

Wi, W2 are resolving sets of G [20].

It is obviously, that W^ W2 = 0. Let W = W^ W2. We differentiate two cases for w G W .If w G Wi, then W2 c W\{w}, if w G W2, then Wi c W\{w}. We can point out that, in any case, W \ {w} is resolving sets of G. Thus W is a fault-tolerant resolving set.

This proves the theorem.

Now we present the proof of the Theorem 3.

Proof. Let n and m be integers, such that m is even, m ^ 2, n ^ 2m — 1 and (m — 1) is a divisor of (n — 1). Let W be a fault-tolerant metric basis of Pm K Pn, and let V be

the vertex set of P„

Pn

We denote к = " . By Lemma 5 we have for all t s {0,..., к — 1}

Vt

t(m-1)+1, (t+1)(m-1)

nw

^ 2.

By Lemma 8 we have

Vn-m+1,nf) W

> 3.

We consider two cases.

Case 1. If there exists t G {0,...,k — 1} such that

Vt

t(m-1)+1, (t+1)(m-1)

since V is the union of sets, that are disjoint,

k-1

W

> 3,

V — U Vt(m-1)+1, (t+1)(m-1) U Vn-m+1, ги

t=0

then

\W\ > 2(k — 1) + 3 + 3 — 2k + 4. Case 2. We assume, that for all t e{0,...,k — 1},

Vt

t(m-1)+1, (t+1)(m-1)

We first consider that t — 0:

V1,m-1f) w

W

— 2.

2.

Then Lemma 7 leads to V1t m-1 p| W с V1, in particular

V1 W

2.

We now take t — 1 in (1) and we get

m, 2(m-1)

W

Since V2, m-1 П W = 0 we can notice, that

V,

2, 2(m-1)

W

— 2.

2.

Then Lemma 7 leads to V2,2(m-1)f]W с Vm, in particular

V2,тГ\ W

2.

Further it is analogically proved by mathematical induction, that

Vt

t(m-1) + 2, (t+1)(m-1)+1

W

— 2

(1)

(2)

(3)

(4)

for all t =1,...,к — 2.

By Lemma 5 we have

V?

(k-1)(m-1) + 2, k(m-1)+1

and

Vk(m-i) + 2, n P| W

Since V is the union of sets, that are disjoint,

^ 2.

^ 2

(5)

(6)

V = U Vt(m-1)+2,

(t+1)(m-1) + 1

О,

t=0

and taking into account the above (2)-(6), we deduce

\Wl > 2k + 4. According to the two cases above we have

n1

f3'(Pm B Pn) = \W\ > 2k + 4 = 2--- + 2.

m—1

By Theorem 2 we have f3'(Pm IE Pn) < 2^ + 2. Hence

n1

l3'(Pm®Pn)= 2--- + 2.

m—1

This proves the theorem.

Figure 2 shows graph P3 K Pii. Vertices of the fault-tolerant metric basis are black,

j3'(P3 K Pii) = 12.

Fig 2. The fault-tolerant metric basis for graph P3 B Pn

Conclusion. Theorem 3 leads to the following inference. The fault-tolerant metric basis for a partcular case of the king's graph contains two times more vertices than the metric basis does. Our conjecture consists of the statement that the upper bound for the fault-tolerant metric dimension of the king's graph from Theorem 2 is an exact value.

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For citation: Voronov R. V. The fault-tolerant metric dimension of the king's graph. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 2017, vol. 13, iss. 3, pp. 241-249. DOI: 10.21638/11701/spbu10.2017.302

Статья рекомендована к печати проф. А. П. Жабко. Статья поступила в редакцию 11 декабря 2016 г. Статья принята к печати 8 июня 2017 г.