Aequationes Math. 71 (2006) 1–180001-9054/06/010001-18DOI 10.1007/s00010-005-2800-z

c© Birkhauser Verlag, Basel, 2006

Aequationes Mathematicae

Research papers

The partition dimension of Cayley digraphs

Melodie Fehr, Shonda Gosselin and Ortrud R. Oellermann∗

Summary. Let G be a (di)graph and S a set of vertices of G. We say S resolves two verticesu and v of G if d(u, S) 6= d(v, S). A partition Π = {P1, P2, . . . , Pk} of V (G) is a resolvingpartition of G if every two vertices of G are resolved by Pi for some i (1 ≤ i ≤ k). The smallestcardinality of a resolving partition of G, denoted by pd(G), is called the partition dimensionof G. A vertex r of G resolves a pair u, v of vertices of G if d(u, r) 6= d(v, r). A set R ofvertices of G is a resolving set for G if every two vertices of G are resolved by some vertex of R.The smallest cardinality of a resolving set of vertices, denoted by dim(G), is called the metricdimension of G. We begin by disproving a conjecture made by Chartrand, Salehi and Zhangregarding the partition dimension of products of graphs. The partition dimension of Cayleydigraphs of abelian groups with a specific minimal set of generators is shown to be at most onemore than the number of generators with equality for one or two generators. It is known thatpd(G) ≤ dim(G)+1. It is pointed out that for every positive integer M there are Cayley digraphsD for which dim(D) − pd(D) ≥ M , and that there are classes of Cayley digraphs D such thatpd(D)dim(D)

→ 0 as |V (D)| → ∞. Moreover, it is shown that the partition dimension of the Cayley

digraph for the dihedral group of order 2n (n ≥ 3) with a minimum set of generators is 3. Weconclude by introducing a more general class of problems for which the problems of finding themetric dimension and partition dimension of a (di)graph are the two extremes and provide aninterpretation of the transition between these two invariants.

Mathematics Subject Classification (2000). 05C12, 05C20, 05C90.

Keywords. Metric dimension, Cayley digraphs, partition dimension.

1. Introduction

Let G be a connected graph. The distance between two vertices u and v of G,denoted by d(u, v), is the length of a shortest u−v path in G. The distance from avertex v of G to a subset S of the vertices of G is defined as min{d(v, x)|x ∈ S}, andis denoted by d(v, S). We say S resolves two vertices u and v if d(u, S) 6= d(v, S).A partition Π = {P1, P2, . . . , Pk} of V (G) is a resolving partition of V (G) if everytwo vertices of G are resolved by Pi for some i (1 ≤ i ≤ k). Alternatively, ifΠ = {P1, P2, . . . , Pk} is a partition of V (G) into k sets that have been assignedthe given order, then Π is a resolving partition if and only if, for every two vertices

∗Supported by an NSERC grant Canada.

2 M. Fehr, Sh. Gosselin and O. R. Oellermann AEM

u and v of G, the k-vectors r(v|Π) = (d(v, P1), d(v, P2), . . . , d(v, Pk)) and r(u|Π) =(d(u, P1), d(u, P2), . . . , d(u, Pk)), called the representations of v and u with respectto Π, are distinct. Note that the only vertices with 0 in the i − th component oftheir representation with respect to Π are the vertices in Pi. Thus, in determiningwhether a partition Π = {P1, P2, . . . , Pk} is a resolving partition, one need onlycheck that all pairs of vertices in Pi have distinct representations with respect toΠ, for all i (1 ≤ i ≤ k). The minimum cardinality of a resolving partition ofG is called the partition dimension of G, and is denoted pd(G). For example,pd(Kp) = p and pd(Km,n) = n if n ≤ m.

The partition dimension was first studied by Chartrand, Salehi, and Zhang in[6] and [7], and later by Chappell, Gimbel and Hartman in [1], to possibly gaininsight into another graph parameter; namely, the ‘metric dimension’. To definethe metric dimension, we need the following terminology. A vertex x resolves twovertices u and v of G if d(u, x) 6= d(v, x). A subset S of V (G) is a resolving set forG if every two distinct vertices of G are resolved by a vertex of S. Alternatively,if S = {x1, x2, . . . , xk} is an ordered set of vertices of G, the representation of avertex v with respect to S is the k-vector r(v|S) = (d(v, x1), d(v, x2), . . . , d(v, xk)).Thus S is a resolving set for G if and only if no two vertices of G have the samerepresentation with respect to S. The minimum cardinality of a resolving set forG is the metric dimension of G, denoted dim(G). A minimum resolving set forG is also called a basis for G. Motivated by the problem of uniquely determiningthe location of an intruder in a network, the concept of the metric dimension ofa graph was introduced by Slater in [16] and [17], and studied independently byHarary and Melter in [12]. It has since been studied further in [2], [4], [5], [8], [9]and [14]. Applications of this invariant to the navigation of robots in networksare discussed in [14] and applications to chemistry are given in [2]. Graph theoryterminology not given here can be found in [3]. For groups and Cayley digraphssee [10].

In [7] it was shown that, for a connected graph G, pd(G) ≤ dim(G) + 1. Thisis easy to see for if S = {x1, x2, . . . , xk} is a basis for G, then the partition Π ={P1, P2, . . . , Pk+1}, described by Pi = {xi} (for 1 ≤ i ≤ k) and Pk+1 = V (G) − S,is certainly a resolving partition of V (G).

The partition and metric dimension are parameters that, at first glance, appearto be quite similar, however, they do not always behave in the same manner. Forexample, in [2] it was shown that for a connected graphH, dim(H)≤dim(H×K2)≤ dim(H) + 1 where H ×K2 is the Cartesian product of H and K2. Chartrand,Salehi and Zhang showed in [6] that pd(H × K2) ≤ pd(H) + 1 and conjecturedthat pd(H) ≤ pd(H × K2). However, we show that this is not the case. Thegraph G = K1,5 × K2 is obtained from two copies, G1 and G2, of K1,5 by joiningcorresponding vertices. Let u1, u2, . . . , u5 and v1, v2, . . . , v5 be the leaves of G1

and G2, respectively, and x and y be their respective centers (see Figure 1).Let Π = {P1, P2, P3, P4}, in which P1 = {u1, u2, u3}, P2 = {v1, v5, y}, P3 =

{u4, v2, x} and P4 = {u5, v3, v4}. Then r(u1|Π) = (0, 1, 1, 2) and r(u2|Π) =

Vol. 71 (2006) The partition dimension of Cayley digraphs 3

Fig. 1. K1,5 × K2

(0, 2, 1, 2), and hence u1 and u2 are resolved by P2. Similarly, all other pairsof distinct vertices of G are resolved by Pi for some i (1 ≤ i ≤ 4), and thus Πis a resolving partition of V (G). Hence pd(K1,5 × K2) ≤ |Π| = 4. In general,a resolving (m − 1)-partition can be found for the graph K1,m × K2, for m ≥ 5.Label the vertices as above where now the leaves of G1 and G2 are ui and vi,respectively, for 1 ≤ i ≤ m. Let Π′ = {P1, . . . , Pm−1}, in which P1, P2, P3, P4 areas above and Pi = {ui+1, vi+1} (for 5 ≤ i ≤ m − 1). It is easy to see that Π′ is aresolving (m− 1)-partition of K1,m ×K2. It was shown in [7] that pd(K1,m) = m,for m ≥ 2. Hence pd(K1,m × K2) ≤ |Π′| ≤ m − 1 < m = pd(K1,m), for m ≥ 5.Thus we have a family of counterexamples to the above conjecture.

The definitions of metric and partition dimension can be extended to directedgraphs. Let D be a directed graph and u, v ∈ V (D). Then the distance from

u to v, denoted by d(u, v), is the number of arcs in a shortest directed u − vpath if one exists and is ∞ otherwise. If S ⊂ V (D), the distance from v to S isdefined by d(v, S) = min{d(v, x)|x ∈ S}. As in the undirected case a vertex x ofD (or a subset S ⊂ V (D)) resolves a pair u, v of vertices of D if d(u, x) 6= d(v, x)(d(u, S) 6= d(v, S), respectively). In [14], Khuller, Raghavachari and Rosenfeldgave a construction that shows that finding the metric dimension of a graph is NP-hard (see also [11]). The metric dimension for trees was established independentlyin [2], [12], [14], and [16]. Some bounds for this invariant, in terms of the diameterof the graph, are given in [2]. However, very little is known about this invariant forgeneral graphs. One may expect a correlation between the automorphism groupof a graph and its metric dimension. However, the results on trees show that thereare trees with arbitrarily large metric dimension having the trivial automorphismgroup. It was also shown in [2] that in general there is no correlation betweenthe metric dimension of a graph and that of its subgraphs. Nevertheless thereappears to be a correlation between the metric dimension and both local andglobal symmetries in (di)graphs that already show higher degrees of symmetry, as

4 M. Fehr, Sh. Gosselin and O. R. Oellermann AEM

for example in vertex transitive (di)graphs.The literature abounds with results on Cayley (di)graphs that depend on their

highly symmetric structure. Cayley (di)graphs with minimal sets of generatorsare also natural models for interconnection networks in computer design as theyrepresent sparse (di)graphs with relatively small diameter. One of the simplestCayley (di)graphs, namely the n-cube has led to a multitude of of deep and inter-esting problems [13]. Bounds for the metric and partition dimension were givenin [1] and [2] and the asymptotically exact value was recently found in [15] to be2n/ log2 n. We explore the metric and partition dimension of Cayley digraphs asthis is a class of vertex transitive graphs for which degrees of symmetry may vary.Cayley digraphs have the added advantage that distances between pairs of verticescan be described algebraically, thus lending themselves more readily to the use ofalgebraic tools when computing ‘distance related’ invariants.

In [9] we presented sharp bounds on the metric dimension of certain typesof Cayley digraphs. In this paper, we reexamine these graphs with respect tothe partition dimension. We close with several open problems. In particular wedefine a general class of problems for which the problems of finding the metricdimension and partition dimension of a graph are the two extreme cases. Thisclass of problems provides a transition from the metric dimension to the partitiondimension of a graph.

2. The partition dimension of Cayley digraphs

The focus of this section is on sharp bounds on the partition dimension of Cayleydigraphs. First, recall the definition of the Cayley digraph for a given group witha specified set of generators (see [10]).

Let Γ be a finite group and ∆ a set of generators for Γ. The Cayley digraph of

Γ with generating set ∆, denoted by Cay(∆ : Γ), is defined as follows:1. The vertices of Cay(∆ : Γ) are precisely the elements of Γ.2. For u and v in Γ, there is an arc from u to v if and only if ug = v for some

generator g ∈ ∆.Note that for a given finite group Γ and a specified set of generators ∆ of

Γ, every element of the group can be expressed as a product of generators in Γ.Hence, in the digraph G = Cay(∆ : Γ), there exists a path from any vertex toevery other vertex. Thus, any Cayley digraph is strongly connected, and both themetric and partition dimensions of any Cayley digraph are therefore defined.

Our analysis of the partition dimension of Cayley digraphs makes use of thefollowing result for strongly connected digraphs.

Proposition 2.1. Let D be a strongly connected digraph. If pd(D) = 2, then

dim(D) ≤ 2.

Vol. 71 (2006) The partition dimension of Cayley digraphs 5

Proof. Let Π = {P1, P2} be a resolving partition of V (D). Since D is stronglyconnected, there is at least one arc directed from P1 into P2 and at least one arcdirected from P2 into P1. Observe that there is at most one vertex of P1 which isthe tail of an arc incident with a vertex of P2, and there is at most one vertex ofP2 which is the tail of an arc incident with a vertex of P1. To see this, supposeto the contrary that v and v′ are distinct vertices of P1 which are each the tail ofsome arc incident with a vertex of P2. Then d(v, P2) = d(v′, P2) = 1, contradictingthe fact that Π is a resolving partition of V (G). Let v be the unique vertex of P1

which is the tail of an arc incident with a vertex of P2, and let w be the uniquevertex of P2 which is the tail of an arc incident with a vertex of P1.

Let B = {v, w}. We claim that B is a resolving set for D. Suppose that x and yare distinct vertices of D. If x and y are both vertices of P1, then d(x, v) 6= d(y, v);otherwise d(x, P2) = d(x, v) + 1 = d(y, v) + 1 = d(y, P2), contradicting the factthat Π is a resolving partition of V (D). Thus v must resolve every pair of distinctvertices in P1. Similarly, w resolves every pair of distinct vertices in P2.

Finally, consider the case in which x and y are from different parts of Π, sayx ∈ P1 and y ∈ P2. If d(x, v) = d(y, v), then necessarily d(x,w) 6= d(y, w), andhence w resolves x and y. This follows from the fact that d(x,w) ≥ d(x, v) + 1 >d(x, v) − 1 = d(y, v) − 1 ≥ d(y, w). Thus every pair of distinct vertices of Dis resolved by a vertex of B, and hence B is a resolving set for D. Thereforedim(D) ≤ |B| = 2. 2

As an example, consider the Cayley digraph D for the symmetric group Γ = S4

with generating set ∆ = {(1, 2, 3, 4), (1, 2)}. The partition dimension of D is 3.Figure 2 shows a resolving 3-partition Π = {P1, P2, P3} of V (D) (the verticesof D are labelled with their representations with respect to this 3-partition); sopd(D) ≤ 3. In [9] it was shown that dim(D) = 3, and hence, by Proposition 2.1,pd(D) ≥ 3.

We turn our attention to the partition dimension of Cayley digraphs for abeliangroups. Since every finite abelian group is isomorphic to a direct product of cyclicgroups, we focus on determining sharp bounds on the partition dimension of theCayley digraph D of the group Γ = Zn1

⊕

Zn2

⊕

. . .⊕

Znkwith generating set

∆ = {(1, 0, . . . , 0), (0, 1, 0, . . . , 0), . . . , (0, 0, . . . , 0, 1)}, in which k, n1, n2, . . . , nk arepositive integers. If k = 1 and n1 ≥ 3, D is the Cayley digraph of a cyclic groupusing one generator, i.e., D is a (nontrivial) directed cycle. It follows from aresult in [4] that the metric dimension of a directed cycle is 1. Thus the partitiondimension of D must be 2. Of course a given abelain group may have more thanone representation as a direct product of cyclic groups. In particular, if ni and nj

are relatively prime, then the two terms Zniand Znj

can be replaced by the singleterm Zninj

to yield a group that is isomorphic to Γ. In this case the generatingset has one less element than before and every vertex in the corresponding Cayleydigraph for Γ has in- and out- degree one less than before.

Suppose now that k ≥ 2. Observe that the vertices of D are k-vectors, and the

6 M. Fehr, Sh. Gosselin and O. R. Oellermann AEM

Fig. 2. A resolving 3-partition of D = Cay({(1, 2, 3, 4), (1, 2)} : S4)

i-th entry of each of these vectors is an integer between 0 and ni−1 (for 1 ≤ i ≤ k).If v = (v1, v2, . . . , vk) and w = (w1, w2, . . . , wk) are vertices of D, one can showthat the distance from v to w in D is given by

d(v, w) =

k∑

i=1

(wi − vi) mod ni.

We now prove that the partition dimension of D is at most k + 1. Theorem 2.4will show that this bound is sharp.

Theorem 2.2. Let k, n1, n2, . . . , nk be positive integers with k ≥ 2. Let G be the

Cayley digraph of the group Zn1

⊕

Zn2

⊕

...⊕

Znkwith generating set {(1, 0, . . . , 0),

(0, 1, 0, . . . , 0), . . . , (0, 0, . . . , 0, 1)}. Then pd(G) ≤ k + 1.

Proof. Let Π = {P1, P2, . . . , Pk+1} in which P1 = {(m, 0, . . . , 0)|0 ≤ m ≤ n1 − 1},Pi = {y = (0, . . . , 0, yi, 0, . . . , 0)|1 ≤ yi ≤ ni − 1} (for 2 ≤ i ≤ k), and Pk+1 =V (G) − (P1 ∪ P2 ∪ . . . ∪ Pk). We show that Π is a resolving partition of V (G).

Vol. 71 (2006) The partition dimension of Cayley digraphs 7

Let v 6= w ∈ P1, say v = (s, 0, . . . , 0) and w = (t, 0, . . . , 0), where 0 ≤ s <t ≤ n1 − 1. Then d(v, P2) = n1 − s + 1 and d(w,P2) = n1 − t + 1. Since s 6= t,d(v, P2) 6= d(w,P2), and thus P2 resolves v and w.

Similarly one can show that if v and w are two vertices of Pi for some i ∈{2, 3, . . . , k}, then P1 resolves v and w.

It remains to show, if v and w are distinct vertices of Pk+1, that they areresolved by Pi for some i (1 ≤ i ≤ k). Suppose, to the contrary, that there existdistinct vertices v and w of Pk+1 which are not resolved by any of the sets Pi.Then d(v, Pi) = d(w,Pi) for all i ∈ {1, . . . , k}. Let v = (v1, v2, . . . , vk) and w =(w1, w2, . . . , wk).

Since d(v, P1) = d(w,P1) and since every vertex x = (x1, x2, . . . , xk) of P1 is ofthe form (m, 0, . . . , 0) (for 0 ≤ m ≤ n1 − 1), we have

min0≤m≤n1−1

{(m − v1) mod n1 + (0 − v2) mod n2 + . . . + (0 − vk) mod nk}

= min0≤m≤n1−1

{(m − w1) mod n1 + (0 − w2) mod n2 + . . . + (0 − wk) mod nk}.

The minimum on the left-hand side of the previous equation occurs when m = v1

and the minimum on the right-hand side occurs when m = w1. Hence

k∑

j=2

vj =k

∑

j=2

wj .

Since d(v, Pi) = d(w,Pi) for all i ∈ {2, . . . , k}, and as every vertex x of Pi hasi-th entry m, for some m ∈ {1, 2, . . . , ni − 1}, and 0 in every entry other than thei-th, it follows that

min1≤m≤ni−1

{(0 − v1) mod n1 + . . . + (m − vi) mod ni + . . . + (0 − vk) mod nk}

= min1≤m≤ni−1

{(0−w1) mod n1+. . .+(m−wi) mod ni+. . .+(0−wk) mod nk}. (1)

There are two cases to consider.

Case 1: For all i (2 ≤ i ≤ k), either vi = wi = 0 or vi 6= 0 and wi 6= 0. Foreach i, if vi 6= 0, then wi 6= 0, and so the minima on the left side and the rightside of equation (1) occur when m = vi and m = wi, respectively. This yields theequation

k∑

j=1j 6=i

vj =

k∑

j=1j 6=i

wj . (2)

If vi = 0 = wi, then both minima in equation (1) occur when m = 1, so thatequation (2) again holds. Hence, in this case, equation (2) holds for all i (1≤ i≤k).This system of equations yields the matrix equation

Av = Aw

8 M. Fehr, Sh. Gosselin and O. R. Oellermann AEM

in which A = [aij ] is the k × k matrix such that

aij =

{

1 if i 6= j0 if i = j.

It can be shown that the matrix A is invertible for all positive integers k ≥ 2 (see[18, p. 169]), so that in this case we must have that v = w, contradicting the factthat v and w are distinct vertices of Pk+1.

Case 2: There is an i ≥ 2 such that vi 6= 0 and wi = 0, or vi = 0 and wi 6= 0.Suppose the latter situation occurs. Then for some i (2 ≤ i ≤ k), the minimumon the left side of equation (1) occurs when m = 1, and the minimum on the rightside of equation (1) occurs when m = wi. This yields the equation

k∑

j=1j 6=i

vj − 1 =k

∑

j=1j 6=i

wj . (3)

Equation (3) holds for all i such that vi = 0 and wi 6= 0. Similarly, for all i suchthat wi = 0 and vi 6= 0, we have

k∑

j=1j 6=i

vj =

k∑

j=1j 6=i

wj − 1. (4)

For all other integers i ∈ {1, . . . , k}, either i = 1 or Case 1 holds, and in each ofthese situations we’ve shown that equation (2) must hold. Using equations (2), (3)and (4), we construct a system of k equations, which yields the matrix equation

Av − z = Aw − y (5)

in which A is the k × k matrix defined in Case 1, z = (z1, z2, . . . , zk), y =(y1, y2, . . . , yk), z1 = y1 = 0, and for 2 ≤ i ≤ k,

zi =

{

1 if vi = 0 and wi 6= 00 otherwise,

and

yi =

{

1 if wi = 0 and vi 6= 00 otherwise.

It can be shown that A−1 = [bij ] is the k × k matrix defined as follows:

bij =

{

1k−1 if i 6= j−(k−2)

k−1 if i = j.

Thusv − A−1z = w − A−1y. (6)

Let a denote the number of places j (2 ≤ j ≤ k) for which vj = 0 and wj 6= 0,and let b denote the number of places j (2 ≤ j ≤ k) for which wj = 0 and vj 6= 0.

Vol. 71 (2006) The partition dimension of Cayley digraphs 9

Since v and w are vertices of Pk+1 they must each contain at least two nonzeroentries, so that a ≤ k − 2 and b ≤ k − 2. By the condition of Case 2, for some i,either vi = 0 and wi 6= 0 or wi = 0 and vi 6= 0. Say the former situation holdsfor i = i′. Then we can be sure that a ≥ 1, and equating the i′-th entries of thek-vectors in equation (6) now yields

vi′ −a

k − 1+ 1 = wi′ −

b

k − 1,

and since vi′ = 0, this implies that

wi′ = 1 −a − b

k − 1. (7)

Since wi′ must be a positive integer and 0 ≤ |a−b| ≤ k−2, equation (7) impliesthat a = b and hence wi′ = 1. (Note that if k = 2, the fact that a = b gives acontradiction to the conditions of Case 2, so we may assume that k ≥ 3.) Sincea ≥ 1 and a = b, b ≥ 1 and so there must be at least one j, say j = j′ (2 ≤ j′ ≤ k),such that j′ 6= i′, wj′ = 0 and vj′ 6= 0. Equating the j′-th entries in the k-vectorsof equation (7) then yields

wj′ −a

k − 1+ 1 = vj′ −

b

k − 1,

which implies that vj′ = 1. Assume, without loss of generality, that i′ = 2 andj′ = 3. Then the following three equations hold:

k∑

j=2

vj =

k∑

j=2

wj (8)

( k∑

j=1j 6=2

vj

)

− 1 =

k∑

j=1j 6=2

wj (9)

k∑

j=1j 6=3

vj =

( k∑

j=1j 6=3

wj

)

− 1. (10)

Equation (8) and the fact that v2 = w3 = 0 and v3 = w2 = 1 then yields

k∑

j=4

vj =

k∑

j=4

wj . (11)

Subtracting equation (11) from equation (10) yields v1 = w1. Hence

v = (c, 0, 1, v4, v5, . . . , vk)

andw = (c, 1, 0, w4, w5, . . . , wk),

10 M. Fehr, Sh. Gosselin and O. R. Oellermann AEM

for some c ∈ {0, 1, . . . , n1 − 1}. Now the assumption that d(v, P2) = d(w,P2)implies that

min1≤m≤n2−1

{

(0−c) mod n1+(m−0) mod n2+(0−1) mod n3+

k∑

j=4

(0−vj) mod nj

}

= min1≤m≤n2−1

{

(0−c) mod n1+(m−1) mod n2+(0−0) mod n3+k

∑

j=4

(0−wj) mod nj

}

.

The minima on both sides of the equation occur when m = 1, which gives

(n1 − c) + 1 + (n3 − 1) +

k∑

j=4

(nj − vj) = (n1 − c) +

k∑

j=4

(nj − wj).

But this implies that

n3 +

k∑

j=4

wj =

k∑

j=4

vj ,

and since n3 ≥ 1,k

∑

j=4

vj >

k∑

j=4

wj ,

which contradicts equation (11). 2

The following result was established in [9].

Theorem 2.3. Let m and n be positive integers. Let D be the Cayley digraph for

the group Zm

⊕

Zn with generating set {(1, 0), (0, 1)}. Then dim(D) = min{m,n}.

We now show that the partition dimension of the same Cayley digraph is only 3.

Theorem 2.4. Let m and n be positive integers greater than 1. Let D be the

Cayley digraph for the group Zm

⊕

Zn with generating set {(1, 0), (0, 1)}. Then

pd(D) = 3.

Proof. The fact that pd(D) ≤ 3 follows directly from Theorem 2.2. It remains toshow that pd(D) ≥ 3. If m ≥ 3 and n ≥ 3, then Theorem 2.3 guarantees thatdim(D) = min{m,n} ≥ 3, and thus, by Proposition 2.1, pd(D) ≥ 3. If m = 2 andn = 2, then D is an undirected 4-cycle, which has partition dimension 3. Finally,consider the case when m ≥ 3 and n = 2, and assume, to the contrary, that thereexists a resolving partition of V (D) with cardinality 2. Let Π = {P1, P2} be sucha partition. Theorem 2.3 guarantees that dim(D) = min{m,n} = 2, so no singlevertex of D constitutes a resolving set for D. Hence |P1| ≥ 2 and |P2| ≥ 2. In anysuch partition, there exists a pair of distinct vertices {v, v′} in P1, each of which

Vol. 71 (2006) The partition dimension of Cayley digraphs 11

is adjacent to a different vertex of P2 in the underlying simple graph. Let w,w′

be distinct vertices of P2 such that v is adjacent to w and v′ is adjacent to w′ inthe underlying simple graph of D. We now consider two cases depending on howthe edges joining the pairs {v, w} and {v′, w′} are oriented in D.

Case 1: Either (v, w), (v′, w′) ∈ E(D) or (w, v), (w′, v′) ∈ E(D). (Note thatthis case includes the situation where at least one of the pairs {v, w} and {v′, w′}is joined by a symmetric pair of arcs.) In the first case d(v, P2) = 1 = d(v′, P2)and in the second case d(w,P1) = 1 = d(w′, P1). Thus either v and v′ or w andw′ are not resolved by Π.

Case 2: Neither the pair {v, w} nor the pair {v′, w′} is joined by a symmetricpair of arcs and either (v, w), (w′, v′) ∈ E(D) or (w, v), (v′, w′) ∈ E(D). Supposethat the former situation occurs. Since D = Cay({(1, 0), (0, 1)} : Zm

⊕

Z2), everyvertex of D is incident with one symmetric edge, so there exist distinct verticesx and y of D, x 6= w and y 6= v, such that the pairs {v, x} and {w, y} are eachjoined by a symmetric pair of arcs in D. Hence D contains the subgraph shown inFigure 3 in which the black vertex v is in P1, the grey vertex w is in P2, and thewhite vertices x and y are as yet unassigned to a set in the partition. If y ∈ P1, yand v are not resolved. Thus y ∈ P2. Now if x ∈ P1, x and v are unresolved. Onthe other hand, if x ∈ P2, then x and w′ are unresolved. Clearly, no 2-partition ofV (D) will be a resolving partition in this case either.

Thus pd(D) ≥ 3. 2

Fig. 3. A subgraph of D

Remark. From Theorems 2.3 and 2.4 it follows that the difference between themetric dimension and partition dimension can be made larger than any positiveinteger M. Moreover, for the class of digraphs D = Cay({(1, 0), (0, 1)} : Zm

⊕

Zm),pd(D)dim(D) → 0 as V (D) → ∞. This is also true for the next class of Cayley digraphs

that we consider.

Let n be a positive integer. Consider the group of symmetries of the regular n-gon, called the dihedral group of order 2n, and denoted by Dn. This group consistsof n rotations and n reflections. Let G be the Cayley digraph for the group Dn withgenerating set {R 360

n, A}, in which A is any reflection in the group and R 360

nis the

12 M. Fehr, Sh. Gosselin and O. R. Oellermann AEM

anticlockwise rotation through an angle of 360n degrees. Then G can be obtained

from two directed cycles, C1 = v1, v2, . . . , vn, v1 and C2 = v′1, v

′n, v′

n−1, . . . , v′1,

by joining vi and v′i by an edge (i.e. a symmetric pair of arcs) for 1 ≤ i ≤ n.

Note that C1 and C2 are oriented in opposite directions. In [9] it was shown thatdim(G = Cay({R 360

n, A} : Dn)) = n, for n ≥ 3. For 3 ≤ n ≤ 11, it can be shown

that pd(G) = 3. In the next result we consider the cases where n ≥ 12.

Theorem 2.5. Let n ≥ 12 be a positive integer and G the Cayley digraph for the

group Dn with generating set {R 360

n, A} as described above. Then pd(G) = 3.

Proof. It was shown in [9] that dim(G) = n for n ≥ 3. Hence Proposition 2.1guarantees that pd(G) ≥ 3. To show that pd(G) ≤ 3, we now describe a resolving3-partition Π = {P1, P2, P3} of V (G). Let n = 4q+r where q and r are nonnegativeintegers, q ≥ 3 and r = 0, 1, 2 or 3. We consider four cases depending on r.

Case 1: r = 0P1 = {v1, v2, v3, v

′1, v

′n−1, v

′n}. (Note that P1 contains exactly three consecutive

vertices from C1 and three consecutive vertices from C2 and the paths induced bythese two sets of vertices are joined by the edge v1v

′1.)

P2 = {v4i+2, v4i+3|1 ≤ i < n/4}∪{vn}∪{v′4i+1, v

′4i+2|1 ≤ i < n/4}. (Note that

P2 contains all vertices vi of C1 where i = n and where i ≡ 2 or 3(mod 4) and allvertices v′

i of C2 where i ≡ 1 or 2(mod 4) except v′1 and v′

2.)

Fig. 4. Cay({R 360

14

, A} : D14)

Vol. 71 (2006) The partition dimension of Cayley digraphs 13

P3 = V (G) − (V (P1) ∪ V (P2)). (Note that P3 contains all vertices vi of C1

where i ≡ 0 or 1(mod 4) except v1 and vn, and it contains v′2 and all vertices v′

i

of C2 where i ≡ 0 or 3(mod 4) except v′n and v′

n−1.)

Case 2: r = 1P1 = {v1, v2, v

′1, v

′n}.

For P2 we consider two cases depending on whether n ≡ 1 or 5(mod 8).n ≡ 1(mod 8): In this case let P2 = {v4i+1, v4i+2|1 ≤ i ≤ ⌊n

8 ⌋ − 1}∪{v4⌊n

8⌋+1}∪{v4i, v4i+1|⌈

n8 ⌉≤ i≤⌊n

4 ⌋}∪{v′4i, v

′4i+1|1 ≤ i ≤ ⌊n

8 ⌋}∪{v′4i+3, v

′4i+4|⌈

n8 ⌉−

1 ≤ i ≤ ⌊n4 ⌋ − 1}

n ≡ 5(mod 8): In this case let P2 = {v4i+1, v4i+2|1 ≤ i ≤ ⌊n8 ⌋} ∪ {v4i, v4i+1|

⌈n8 ⌉ ≤ i ≤ ⌊n

4 ⌋} ∪ {v′4i, v

′4i+1|1 ≤ i ≤ ⌊n

8 ⌋} ∪ {v′4⌈n

8⌉} ∪ {v′

4i+3, v′4i+4|⌈

n8 ⌉ ≤ i ≤

⌊n4 ⌋ − 1}.

P3 = V (G) − (P1 ∪ P2).

Case 3: r = 2P1 = {v1, v2, v

′1, v

′n}.

P2 = {v4i+1, v4i+2|1 ≤ i ≤ ⌊n4 ⌋} ∪ {v′

4i, v′4i+1|1 ≤ i ≤ ⌊n

4 ⌋}.

P3 = V (G) − (P1 ∪ P2).Figure 4 illustrates the partition for n = 14.

Fig. 5. Cay({R 360

15

, A} : D15)

14 M. Fehr, Sh. Gosselin and O. R. Oellermann AEM

Case 4: r = 3P1 = {v1, v2, v3, v

′1, v

′n−1, v

′n}.

For P2 we consider two cases depending on whether n ≡ 3 or 7(mod 8).n ≡ 3(mod 8): In this case let P2 = {v4i+2, v4i+3|1 ≤ i ≤ ⌊n

8 ⌋ − 1} ∪{v4⌊n

8⌋+2} ∪ {v4i+1, v4i+2|⌈

n8 ⌉ ≤ i ≤ ⌊n

4 ⌋} ∪ {vn} ∪ {v′4i+1, v

′4i+2|1 ≤ i ≤ ⌊n

8 ⌋} ∪{v′

4i, v′4i+1|⌈

n8 ⌉ ≤ i ≤ ⌊n

4 ⌋}.n ≡ 7(mod 8): In this case let P2 = {v4i+2, v4i+3|1 ≤ i ≤ ⌊n

8 ⌋} ∪ {v4i+1,v4i+2|⌈

n8 ⌉ ≤ i ≤ ⌊n

4 ⌋}∪{vn}∪{v′4i+1, v

′4i+2|1 ≤ i ≤ ⌊n

8 ⌋}∪{v′4⌈n

8⌉+1}∪{v′

4i, v′4i+1|

⌈n8 ⌉ + 1 ≤ i ≤ ⌊n

4 ⌋}.

P3 = V (G) − (P1 ∪ P2).Figure 5 illustrates the partition for n = 15.In all four cases it is not difficult to see that every two vertices of P1 are resolved

by P2 or P3.To see that every two vertices of Pi are resolved for i = 2, 3, we need only show

that if u, v are two vertices of P2 (respectively, P3) that have the same distanceto P1, then they have different distances to P3 (respectively, P2). We begin byobserving that for every integer d, 1 ≤ d ≤ n/2, there are at most two vertices ofPi that have distance d to P1. Also no vertex of Pi has distance more than n/2to P1, for i = 2, 3. Moreover if two such vertices of P2 (respectively, P3) have thesame distance to P1, then either, one of these is distance 1 and the other distance2 or, one is distance 3 and the other distance 4, to P3 (respectively, P2). 2

3. Concluding remarks and open problems

The dimension and partition dimension of a group

For a group Γ the dimension (partition dimension) of Γ, denoted by dim(Γ) (andpd(Γ), respectively) is defined to be the smallest dimension (partition dimension,respectively) among all Cayley digraphs Cay(∆,Γ) where ∆ is taken over all gen-erating sets for Γ. If Γ is cyclic, then it is not difficult to see that dim(Γ) = 1and pd(Γ) = 2. If Γ is not cyclic, but can be expressed as a direct product oftwo cyclic groups, then every generating set for Γ has at least two elements. Thusevery vertex in a Cayley digraph for Γ has in- and out-degree at least 2. We knowfrom Theorem 2.2 that 2 ≤ pd(Γ) ≤ 3. If there is a resolving partition for someCayley digraph D of Γ that has exactly two elements, say P1, P2, then every twovertices of P1 are different distances from P2. Let v1 be such that d(v1, P2) = 1.If v1 is the only vertex of P1, then any two of its in-neighbours are in P2 andthus not resolved by the partition {P1, P2} of V (D). So we may assume that somein-neighbour v2 of v1 belongs to P2. But then both v1 and v2 have an in-neighbourthat belongs to P2. One can choose these two in-neighbours, say v′

1 and v′2 (of

v1 and v2, respectively), to be distinct. So {P1, P2} does not resolve D. Hencepd(Γ) = 3. If Γ is not cyclic but can be expressed as a direct product of two cyclic

Vol. 71 (2006) The partition dimension of Cayley digraphs 15

groups, then Γ = Zm

⊕

Zn where m is as small as possible. In this case one canshow, using the techniques employed in [9], that dim(Γ) = m. In general howeverthe problem of finding the dimension and partition dimension of a group remainsopen.

The l-partition dimension of a graph

We now describe a class of problems for which the problems of finding the metricdimension and partition dimension of a graph are the two extremes. Let G bea (di)graph of order p ≥ 2 and l a positive integer (1 ≤ l ≤ ⌈p/2⌉). Then thel-partition dimension of G, denoted by pdl(G), is the smallest k for which there isa resolving partition Π = {P1, P2, . . . , Pk} of V (G) such that |Pi| ≤ l for 1 ≤ i ≤k − 1. We call such a resolving partition an l-resolving partition. If l = 1, the setP1 ∪P2 ∪ . . .∪Pk−1 is a minimum resolving set for G; thus dim(G) = pd1(G)− 1.Moreover, pd(G) = pd⌈p/2⌉(G). It is not difficult to see that pd1(G) ≥ pd2(G) ≥. . . ≥ pd⌈p/2⌉(G). Thus the sequence pd1(G),pd2(G), . . . ,pd⌈p/2⌉(G), called thedimension transition sequence of G, is a non-increasing sequence of integers. Whileit is not the main object of this paper to study dimension transition sequences ofgraphs, the next result shows how this sequence behaves in the case of the Cayleydigraph for the dihedral group Dn of order p = 2n.

Theorem 3.1. 1) pdl(Cay({R 360

n, A} : Dn)) = ⌈n

l ⌉ + 1 for 1 ≤ l < n/2; and

2) 3 ≤ pdl(Cay({R 360

n, A} : Dn)) ≤ 4 for n/2 ≤ l ≤ n.

Proof. 1) For the Cayley digraph G of the dihedral group Dn of order 2n with aminimum set of generators, the l-partition dimension can easily be found if l < n/2as we now see. A pair of vertices in a digraph are redundant if they have the sameout-neighbourhood. It is not difficult to see that the vertices of this Cayley digraphcan be partitioned into n redundant pairs. Since vertices from a redundant paircannot belong to the same element of a resolving partition Π = {P1, P2, . . . , Pk},|Pk| ≤ n and so k ≥ ⌈n

l ⌉ + 1. Suppose n = lq + r for 0 ≤ r < l. Let C1 and C2

be the cycles of the Cayley digraph for the dihedral group of order 2n describedprior to Theorem 2.5. Let Pi = {v(i−1)l+t|1 ≤ t ≤ l} for 1 ≤ i ≤ q and if r 6= 0let Pq+1 = V (C1) − {P1 ∪ P2 ∪ . . . ∪ Pq}. Finally let P⌈n

l⌉+1 = V (C2). Then it is

not difficult to see that Π = {P1, P2, . . . , P⌈nl⌉+1} is a resolving partition. Hence

pdl(G) = ⌈nl ⌉ + 1. (Note that ⌈n

l ⌉ + 1 ≥ 4 for 1 ≤ l < n/2.)2) By Theorem 2.5, pd(Cay({R 360

n, A} : Dn)) = 3, and thus pdl(Cay({R 360

n, A} :

Dn)) ≥ 3, for n/2 ≤ l ≤ n. To complete the proof we show if n/2 ≤ l ≤ n,that pdl(Cay({R 360

n, A} : Dn)) ≤ 4. Let n = 4q + r where q and r are non-

negative integers and r = 0, 1, 2 or 3. We now describe a resolving 4-partitionΠ = {P1, P2, P3, P4} of V (G) with the property that |Pi| ≤ ⌈n

2 ⌉ ≤ l for 1 ≤ i ≤ 3.We consider four cases that depend on r.

16 M. Fehr, Sh. Gosselin and O. R. Oellermann AEM

Case 1: r = 0P1 = {v1, v2, . . . , vq} ∪ {v′

1, v′n, v′

n−1, . . . , v′n−q+2},

P2 = {vq+1, vq+2, . . . , v2q} ∪ {v′q+1, v

′q, . . . , v

′2},

P3 = {v2q+1, v2q+2, . . . , v3q} ∪ {v′2q+1, v

′2q, . . . , v

′q+2},

P4 = {v3q+1, v3q+2, . . . , vn} ∪ {v′3q+1, v

′3q, . . . , v

′2q+2}.

Case 2: r = 1P1 = {v1, v2, . . . , vq+1} ∪ {v′

1, v′n, v′

n−1, . . . , v′n−q+2},

P2 = {vq+2, vq+3, . . . , v2q+1} ∪ {v′q+2, v

′q+1, . . . , v

′2},

P3 = {v2q+2, v2q+3, . . . , v3q+1} ∪ {v′2q+2, v

′2q+1, . . . , v

′q+3},

P4 = {v3q+2, v3q+3, . . . , vn} ∪ {v′3q+2, v

′3q+1, . . . , v

′2q+3}.

Case 3: r = 2P1 = {v1, v2, . . . , vq+1} ∪ {v′

1, v′n, v′

n−1, . . . , v′n−q+2},

P2 = {vq+2, vq+3, . . . , v2q+1} ∪ {v′q+2, v

′q+1, . . . , v

′2},

P3 = {v2q+2, v2q+3, . . . , v3q+2} ∪ {v′2q+2, v

′2q+1, . . . , v

′q+3},

P4 = {v3q+3, v3q+4, . . . , vn} ∪ {v′3q+3, v

′3q+2, . . . , v

′2q+3}.

Case 4: r = 3P1 = {v1, v2, . . . , vq+1} ∪ {v′

1, v′n, v′

n−1, . . . , v′n−q+2},

P2 = {vq+2, vq+3, . . . , v2q+2} ∪ {v′q+2, v

′q+1, . . . , v

′2},

P3 = {v2q+3, v2q+4, . . . , v3q+3} ∪ {v′2q+3, v

′2q+2, . . . , v

′q+3},

P4 = {v3q+4, v3q+5, . . . , vn} ∪ {v′3q+4, v

′3q+3, . . . , v

′2q+4}.

Figure 6 illustrates the partition for n = 14.

Fig. 6. Cay({R 360

14

, A} : D14)

Vol. 71 (2006) The partition dimension of Cayley digraphs 17

It can be shown in each of the four cases that the given partition is resolving.Thus pdl(G) ≤ 4. 2

Remark. From the proof of Theorem 2.5 we see that the lower bound of Theo-rem 3.1 (2) is attained for l = n, n − 1 and n − 2, and if n ≡ 0 or 3(mod 4), thislower bound is also attained for l = n − 3.

Studying the behaviour of dimension transition sequences for graphs in generalremains an open problem. It may also be an interesting problem to impose condi-tions on the elements of a resolving partition such as that of requiring that theybe independent sets.

References

[1] G. Chappell, J. Gimbel and C. Hartman, Bounds on the metric and partition dimensionsof a graph. Preprint.

[2] G. Chartrand, L. Eroh, M. Johnson and O. R. Oellermann, Resolvability in graphs

and the metric dimension of a graph, Discrete Appl. Math. 105 (2000), 99–113.

[3] G. Chartrand and L. Lesniak, Graphs and Digraphs, Third Edition, Chapman and Hall,New York, 1996.

[4] G. Chartrand, M. Raines and P. Zhang, The directed distance dimension of oriented

graphs, Math. Bohem. 125 (2000), 155–168.

[5] G. Chartrand, M. Raines and P. Zhang, On the dimension of oriented graphs, Util.Math. 60 (2001), 139–151.

[6] G. Chartrand, E. Salehi and P. Zhang, On the partition dimension of a graph, Congr.Numer. 130 (1998), 157–168.

[7] G. Chartrand, E. Salehi and P. Zhang, The partition dimension of a graph, AequationesMath. 59 (2000), 45–54.

[8] J. Currie and O. R. Oellermann, The metric dimension and metric independence of a

graph, J. Combin. Math. Combin. Comput. 39 (2001), 157–167.

[9] M. Fehr, S. Gosselin and O. Oellermann, The metric dimension of Cayley digraphs,Preprint.

[10] J. A. Gallian, Contemporary Abstract Algebra, Houghton Mifflin Company, New York,2002.

[11] M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory

of NP-Completeness, Freeman, New York, 1979.

[12] F. Harary and R. A. Melter, On the metric dimension of a graph, Ars Combin. 2 (1976),191–195.

[13] W. Imrich and S. Klavzar, Product Graphs: Structure and Recognition, Wiley, New York,2000.

[14] S. Khuller, B. Raghavachari and A. Rosenfeld, Localization in graphs, TechnicalReport (1994).

[15] A. Sebo and E. Tannier, On metric generators of graphs, Math. Oper. Res. 29 (2) (2004),383–393.

[16] P. J. Slater, Leaves of trees, Congr. Numer. 14 (1975), 549–559.

[17] P. J. Slater, Dominating and reference sets in a graph, J. Math. Phys. Sci. 22 (1988),445–455.

[18] G. Strang, Linear Algebra and its Application, Second Ed., Academic Press, New York,1980.

18 M. Fehr, Sh. Gosselin and O. R. Oellermann AEM

Melodie Fehr, Shonda Gosselin and Ortrud R. OellermannThe University of Winnipeg515 Portage AvenueWinnipeg, MB R3B 2E9Canadae-mail: o.oellermann@uwinnipeg.ca

Manuscript received: September 1, 2003 and, in final form, May 15, 2004.