On the partition dimension of trees

Abstract

Given an ordered partition $\Pi =\{P_1,P_2, ...,P_t\}$ of the vertex set $V$
of a connected graph $G=(V,E)$, the \emph{partition representation} of a vertex
$v\in V$ with respect to the partition $\Pi$ is the vector
$r(v|\Pi)=(d(v,P_1),d(v,P_2),...,d(v,P_t))$, where $d(v,P_i)$ represents the
distance between the vertex $v$ and the set $P_i$. A partition $\Pi$ of $V$ is
a \emph{resolving partition} of $G$ if different vertices of $G$ have different
partition representations, i.e., for every pair of vertices $u,v\in V$,
$r(u|\Pi)\ne r(v|\Pi)$. The \emph{partition dimension} of $G$ is the minimum
number of sets in any resolving partition of $G$. In this paper we obtain
several tight bounds on the partition dimension of trees.

Full text

Page 1

arXiv:1110.5289v1 [math.CO] 24 Oct 2011
On the partition dimension of trees
Juan A. Rodr´ ıguez-Vel´ azquez1, Ismael G. Yero1and
Magdalena Lema´ nska2
1Departament d’Enginyeria Inform` atica i Matem` atiques
Universitat Rovira i Virgili, Av. Pa¨ ısos Catalans 26, 43007 Tarragona, Spain.
ismael.gonzalez@urv.cat, juanalberto.rodriguez@urv.cat
2Department of Technical Physics and Applied Mathematics
Gdansk University of Technology, ul. Narutowicza 11/12 80-233 Gdansk, Poland
magda@mifgate.mif.pg.gda.pl
October 25, 2011
Abstract
Given an ordered partition Π = {P1,P2,...,Pt} of the vertex set
V of a connected graph G = (V,E), the partition representation
of a vertex v ∈ V with respect to the partition Π is the vector
r(v|Π) = (d(v,P1),d(v,P2),...,d(v,Pt)), where d(v,Pi) represents the
distance between the vertex v and the set Pi. A partition Π of V
is a resolving partition of G if different vertices of G have different
partition representations, i.e., for every pair of vertices u,v ∈ V ,
r(u|Π) ?= r(v|Π). The partition dimension of G is the minimum num-
ber of sets in any resolving partition of G. In this paper we obtain
several tight bounds on the partition dimension of trees.
Keywords: Resolving sets, resolving partition, partition dimension.
AMS Subject Classification numbers: 05C12
1

Page 2

1 Introduction
The concepts of resolvability and location in graphs were described indepen-
dently by Harary and Melter [9] and Slater [17], to define the same structure
in a graph. After these papers were published several authors developed di-
verse theoretical works about this topic [2, 3, 4, 5, 6, 7, 8, 9, 10, 14]. Slater
described the usefulness of these ideas into long range aids to navigation
[17]. Also, these concepts have some applications in chemistry for repre-
senting chemical compounds [12, 13] or to problems of pattern recognition
and image processing, some of which involve the use of hierarchical data
structures [15]. Other applications of this concept to navigation of robots
in networks and other areas appear in [5, 11, 14]. Some variations on re-
solvability or location have been appearing in the literature, like those about
conditional resolvability [16], locating domination [10], resolving domination
[1] and resolving partitions [4, 7, 8].
Given a graph G = (V,E) and a set of vertices S = {v1,v2,...,vk} of G,
the metric representation of a vertex v ∈ V with respect to S is the vector
r(v|S) = (d(v,v1),d(v,v2),...,d(v,vk)), where d(v,vi) denotes the distance
between the vertices v and vi, 1 ≤ i ≤ k. We say that S is a resolving set
of G if different vertices of G have different metric representations, i.e., for
every pair of vertices u,v ∈ V , r(u|S) ?= r(v|S). The metric dimension1of
G is the minimum cardinality of any resolving set of G, and it is denoted by
dim(G). The metric dimension of graphs is studied in [2, 3, 4, 5, 6, 18].
Given an ordered partition Π = {P1,P2,...,Pt} of the vertices of G, the
partition representation of a vertex v ∈ V with respect to the partition Π
is the vector r(v|Π) = (d(v,P1),d(v,P2),...,d(v,Pt)), where d(v,Pi), with
1 ≤ i ≤ t, represents the distance between the vertex v and the set Pi, i.e.,
d(v,Pi) = minu∈Pi{d(v,u)}. We say that Π is a resolving partition of G if
different vertices of G have different partition representations, i.e., for every
pair of vertices u,v ∈ V , r(u|Π) ?= r(v|Π). The partition dimension of G is
the minimum number of sets in any resolving partition of G and it is denoted
by pd(G). The partition dimension of graphs is studied in [4, 7, 8, 18].
1Also called locating number.
2

Page 3

2 The partition dimension of trees
It is natural to think that the partition dimension and metric dimension are
related; in [7] it was shown that for any nontrivial connected graph G we
have
pd(G) ≤ dim(G) + 1. (1)
We know that the partition dimension of any path is two.
any path graph P, it follows pd(P) = dim(P) + 1 = 2. A formula for the
dimension of trees that are not paths has been established in [5, 9, 17]. In
order to present this formula, we need additional definitions. A vertex of
degree at least 3 in a tree T will be called a major vertex of T. Any leaf u of
T is said to be a terminal vertex of a major vertex v of T if d(u,v) < d(u,w)
for every other major vertex w of T. The terminal degree of a major vertex v
is the number of terminal vertices of v. A major vertex v of T is an exterior
major vertex of T if it has positive terminal degree.
That is, for
Figure 1: In this tree the vertex 3 is an exterior major vertex of terminal
degree two: 1 and 4 are terminal vertices of 3.
Let n1(T) denote the number of leaves of T, and let ex(T) denote the
number of exterior major vertices of T. We can now state the formula for
the dimension of a tree [5, 9, 17]: if T is a tree that is not a path, then
dim(T) = n1(T) − ex(T). (2)
As a consequence, if T is a tree that is not a path, then
pd(T) ≤ n1(T) − ex(T) + 1.(3)
The above bound is tight, it is achieved for the graph in Figure 1 where
Π = {{8},{4,9},{1,2,3,5,6,7}} is a resolving partition and pd(T) = 3.
3

Page 4

Figure 2: Π = {{1,4,9,12},{3,5,8,11},{2,6,7,10}}is a resolving partition.
However, there are graphs for which the following bound gives better result
than bound (3), for instance, the graph in Figure 2.
Let S = {s1,s2,...,sκ} be the set of exterior major vertices of T =
(V,E) with terminal degree greater than one, let {si1,si2,...,sili} be the set
of terminal vertices of siand let τ = max1≤i≤κ{li}. With the above notation
we have the following result.
Theorem 1. For any tree T which is not a path,
pd(T) ≤ κ + τ − 1.
Proof. For a terminal vertex sijof a major vertex si∈ S we denote by Sij
the set of vertices of T, different from si, belonging to the si− sijpath. If
li < τ − 1, we assume Sij = ∅ for every j ∈ {li+ 1,...,τ − 1}. Now for
every j ∈ {2,...,τ − 1}, let Bj = ∪κ
Ai= Si1. Let us show that Π = {A,A1,A2,...Aκ,B2,...,Bτ−1} is a resolving
partition of T, where A = V −?(∪κ
different vertices x,y ∈ V . Note that if x and y belong to different sets of Π,
we have r(x|Π) ?= r(y|Π).
Case 1: x,y ∈ Sij. In this case, d(x,A) = d(x,si) ?= d(y,si) = d(y,A).
Case 2: x ∈ Sij and y ∈ Skl, i ?= k. If j = 1 or l = 1, then x and y
belong to different sets of Π. So we suppose j ?= 1 and l ?= 1. Hence, if
i=1Sij and, for every i ∈ {1,...,κ}, let
i=1Ai) ∪?∪τ−1
j=2Bj
??. We consider two
4

Page 5

d(x,Ai) = d(y,Ai), then
d(x,Ak) = d(x,si) + d(si,sk) + 1
= d(x,Ai) + d(si,sk)
= d(y,Ai) + d(si,sk)
= d(y,sk) + 2d(sk,si) + 1
= d(y,Ak) + 2d(sk,si)
> d(y,Ak).
Case 3: x ∈ Siτ and y ∈ A − ∪κ
d(x,si) = d(y,si). Since y / ∈ Slτ, l ∈ {1,...,κ}, there exists Aj∈ Π, j ?= i,
such that sidoes not belong to the y−sjpath. Now let Y be the set of vertices
belonging to the y−sjpath, and let v ∈ Y such that d(si,v) = min
Hence,
l=1Slτ. If d(x,Ai) = d(y,Ai), then
u∈Y{d(si,u)}.
d(x,Aj) = d(x,si) + d(si,v) + d(v,sj) + 1
= d(y,si) + d(si,v) + d(v,sj) + 1
= d(y,v) + 2d(v,si) + d(v,sj) + 1
= d(y,Aj) + 2d(v,si)
> d(y,Aj).
Case 4: x,y ∈ A′= A−∪κ
the vertex x belongs to the y − sipath or the vertex y belongs to the x − si
path, then d(x,Ai) ?= d(y,Ai). Otherwise, there exist at least two exterior
major vertices si, sjsuch that the x−y path and the si−sjpath share more
than one vertex (if not, then x,y / ∈ A′). Let W be the set of vertices belonging
to the si− sjpath. Let u,v ∈ W such that d(x,u) = minz∈W{d(x,z)} and
d(y,v) = minz∈W{d(y,z)}. We suppose, without loss of generality, that
d(si,u) > d(v,si). Hence, if d(x,v) = d(y,v), then d(x,u) ?= d(y,u), and if
d(x,u) = d(y,u), then d(x,v) ?= d(y,v). We have,
l=1Slτ. If for some exterior major vertex si∈ S,
d(x,Aj) = d(x,u) + d(u,sj) + 1
?= d(y,u) + d(u,sj) + 1
= d(y,Aj)
5

Page 6

or
d(x,Ai) = d(x,v) + d(v,si) + 1
?= d(y,v) + d(v,si) + 1
= d(y,Ai).
Therefore, for different vertices x,y ∈ V, we have r(x|Π) ?= r(y|Π).
One example where pd(T) = κ + τ − 1 is the tree in Figure 1.
Any vertex adjacent to a leaf of a tree T is called a support vertex. In the
following result ξ denotes the number of support vertices of T and θ denotes
the maximum number of leaves adjacent to a support vertex of T.
Corollary 2. For any tree T of order n ≥ 2, pd(T) ≤ ξ + θ − 1.
Proof. If T is a path, then ξ = 2 and θ = 1, so the result follows. Now we
suppose T is not a path. Let v be an exterior major vertex of terminal degree
τ. Let x be the number of leaves adjacent to v and let y = τ − x. Since
κ + y ≤ ξ and x ≤ θ, we deduce κ + τ ≤ ξ + θ.
The above bound is achieved, for instance, for the graph of order six
composed by two support vertices a and b, where a is adjacent to b, and four
leaves; two of them are adjacent to a and the other two leaves are adjacent
to b. One example of graph for which Theorem 1 gives better result than
Corollary 2 is the graph in Figure 1.
Since the number of leaves, n1(T), of a tree T is bounded below by
ξ + θ − 1, Corollary 2 leads to the following bound.
Remark 3. For any tree T of order n ≥ 2, pd(T) ≤ n1(T).
Now we are going to characterize all the trees for which pd(T) = n1(T).
It was shown in [7] that pd(G) = 2 if and only if the graph G is a path. So
by the above remark we obtain the following result.
Remark 4. Let T be a tree of order n ≥ 4. If n1(T) = 3, then pd(T) = 3.
Theorem 5. Let T be a tree with n1(T) ≥ 4. Then pd(T) = n1(T) if and
only if T is the star graph.
6

Page 7

Proof.
T = (V,E) ?= Sn, such that pd(T) = n1(T) ≥ 4. Note that by (3) we have
ex(T) = 1. Let t = n1(T) and let Ω = {u1,u2,...,ut} be the set of leaves of T.
Let u ∈ V be the unique exterior major vertex of T. Let us suppose, without
loss of generality, utis a leaf of T such that d(ut,u) = maxui∈Ω{d(ui,u)}.
For the leaves u1,u2,ut ∈ Ω let the paths P = uut1ut2...utrtut, Q =
uu11u12...u1r1u1and R = uu21u22...u2r2u2. Now, let us form the partition Π =
{A1,A2,...,At−2,A}, such that A1= {u11,u12,...,u1r1,u1,ut2,ut3,...,utrt,ut},
A2 = {u21,u22,...,u2r2,u2,ut1}, Ai = {ui}, i ∈ {3,..,t − 2} and A = V −
∪t−2
following cases,
Case 1: x,y ∈ A1. Let us suppose x ∈ P and y ∈ Q. If d(x,A2) =
d(y,A2), then we have
If T = Snis a star graph, it is clear that pd(T) = n1(T). Now, let
i=1Ai. Let us consider two different vertices x,y ∈ V . Hence, we have the
d(x,A) =d(x,ut1) + 1
=d(x,A2) + 1
=d(y,A2) + 1
=d(y,A) + 2
>d(y,A).
Now, if x,y ∈ P or x,y ∈ Q, then d(x,A) ?= d(y,A).
Case 2: x,y ∈ A2. If x = ut1or y = ut1, then let us suppose for instance,
x = ut1, so we have d(x,A1) = 1 < 2 ≤ d(y,A1). On the contrary, if x,y ∈ R,
then d(x,A) ?= d(y,A).
Case 3: x,y ∈ A. If d(x,A1) = d(y,A1), then t ≥ 5 and there exists a
leaf ui, i ?= 1,2,t − 1,t, such that d(x,Ai) = d(x,ui) ?= d(y,ui) = d(y,Ai).
Therefore, for different vertices x,y ∈ V we have r(x|Π) ?= r(y|Π) and Π
is a resolving partition in T, a contradiction.
Figure 3: A Comet graph where 3 = θ = Pd(T).
7

Page 8

Let T be the Comet graph showed in Figure 3. A resolving partition for
T is Π = {A1,A2,A3}, where A1= {x,t}, A2= {y,z} and A3= {u,w}. In
this case, Pd(T) = 3θ.
Remark 6. For any tree T of order n ≥ 2, pd(T) ≥ θ.
Proof. Since different leaves adjacent to the same support vertex must belong
to different sets of a resolving partition, the result follows.
Other examples where pd(T) = θ are the star graphs and the graph in
Figure 2.
Theorem 7. Let T be a tree. If every vertex belonging to the path between
two exterior major vertices of terminal degree greater than one is an exterior
major vertex of terminal degree greater than one, then
pd(T) ≤ max{κ,τ + 1}.
Proof. If T is a path, then τ = 2 and κ = 1, so the result follows. We suppose
T = (V,E) is not a path. Let S = {s1,s2,...,sκ} be the set of exterior
major vertices of T with terminal degree greater than one and let Bi= {si},
i = 1,...,κ. If κ < τ + 1, then for i ∈ {κ + 1,...,τ + 1} we assume Bi= ∅.
Let libe the terminal degree of si, i ∈ {1,...,κ}. If li< i, then we denote by
{si1,...,sili} the set of terminal vertices of si. On the contrary, if li≥ i, then
the set of terminal vertices of siis denoted by {si1,...,sii−1,sii+1,...,sili+1}.
Also, for a terminal vertex sijof a major vertex siwe denote by Sijthe set
of vertices of T, different from si, belonging to the si− sijpath. Moreover,
we assume Sij = ∅ for the following three cases: (1) i = j, (2) i ≤ li< τ
and j ∈ {li+ 2,...,τ + 1}, and (3) i > liand j ∈ {li+ 1,...,τ + 1}. Now,
let t = max{κ,τ + 1} and let Π = {A1,A2,...,At} be composed by the sets
Ai= Bi∪?∪κ
between two exterior major vertices of terminal degree greater than one, is
an exterior major vertex of terminal degree greater than one, then Π is a
partition of V .
Let us show that Π is a resolving partition. Let x,y ∈ V be different
vertices of T. If x,y ∈ Ai, we have the following three cases.
Case 1: x,y ∈ Sji. In this case d(x,Aj) = d(x,sj) ?= d(y,sj) = d(y,Aj).
Case 2: x ∈ Sjiand y ∈ Ski, j ?= k. If d(x,Ak) = d(y,Ak) we have d(y,Aj) >
d(y,sk) = d(y,Ak) = d(x,Ak) > d(x,sj) = d(x,Aj).
Case 3: x = siand y ∈ Sji. As sihas at least two terminal vertices, there
j=1Sji
?, i = 1,...,t. Since every vertex belonging to the path
8

Page 9

exists a terminal vertex silof si, l ?= j, such that d(x,Al) = d(x,Sil) = 1.
Hence, d(y,Al) > d(y,sj) ≥ 1 = d(x,Al). Therefore, for different vertices
x,y ∈ V , we have r(x|Π) ?= r(y|Π).
The above bound is achieved, for instance, for the graph in Figure 4.
Figure 4: Π = {{1,8,11,14},{2,5,12,15},{3,6,9,16},{4,7,10,13}} is a re-
solving partition.
3 On the partition dimension of generalized
trees
A cut vertex in a graph is a vertex whose removal increases the number of
components of the graph and an extreme vertex is a vertex such that its
closed neighborhood forms a complete graph. Also, a block is a maximal
biconnected subgraph of the graph. Now, let F be the family of sequences
of connected graphs G1,G2,...,Gk, k ≥ 2, such that G1is a complete graph
Kn1, n1≥ 2, and Gi, i ≥ 2, is obtained recursively from Gi−1by adding a
complete graph Kni, ni≥ 2, and identifying a vertex of Gi−1with a vertex
in Kni.
From this point we will say that a connected graph G is a generalized
tree if and only if there exists a sequence {G1,G2,...,Gk} ∈ F such that
Gk= G for some k ≥ 2. Notice that in these generalized trees every vertex
is either, a cut vertex or an extreme vertex. Also, every complete graph used
to obtain the generalized tree is a block of the graph. Note that if every
Giis isomorphic to K2, then Gk is a tree, thus justifying the terminology
used. In this section we will be centered in the study of partition dimension
of generalized trees.
Let G = (V,E) be a generalized tree and let R1,R2,...,Rkbe the blocks
of G. A cut vertex v ∈ V is a support cut vertex if there is at least a
9

Page 10

Figure 5: Π = {{4},{7},{10},{5,8,11},{1,2,3,6,9,12}} is a resolving par-
tition for the generalized tree.
block Riof G, in which v is the unique cut vertex belonging to the block
Ri. An extreme vertex is an exterior extreme vertex if it is adjacent to only
one cut vertex. Let S = {s1,s2,...,sζ} be the set of support cut vertices of
G and let {si1,si2,...,sili} be the set of exterior extreme vertices adjacent
to si∈ S. Also, let Q = {Q1,Q2,...,Qϑ} be the set of blocks of G which
contain more than one cut vertex and more than one extreme vertex and let
{qi1,qi2,...,qiti} be the set of extreme vertices belonging to Qi∈ Q. Now, let
φ =max
1≤i≤ζ,1≤j≤ϑ{li,tj}. With the above notation we have the following result.
Theorem 8. For any generalized tree G,
pd(G) ≤
?ζ + ϑ + φ − 1,
ζ + ϑ + 1,
if φ ≥ 3;
if φ ≤ 2.
Proof. For each support cut vertex si∈ S, let Ai= {si1} and for each block
Qj∈ Q, let Bj= {qj1}. Let us suppose φ ≥ 3. For every j ∈ {2,...,li} we
take Mij = {sij} and, if li< φ − 1, then for every j ∈ {li+1,...,φ − 1} we
consider Mij= ∅. Analogously, for every j ∈ {2,...,ti} we take Nij= {qij}
and, if ti< φ − 1, then for every j ∈ {ti+1,...,φ − 1} we consider Nij= ∅.
Now, let Cj=?max{ζ,ϑ}
Let us prove that Π = {A,A1,A2,...Aζ,B1,B2,...,Bϑ,C2,C3,...,Cφ−1} is
a resolving partition of G, where A = V − ∪ζ
begin with, let x,y be two different vertices of G. We have the following
cases.
Case 1: x is a cut vertex or y is a cut vertex. Let us suppose, for instance,
x is a cut vertex. So there exists an extreme vertex si1such that x belongs
to a shortest y − si1path or y belongs to a shortest x − si1path. Hence, we
have d(x,Ai) = d(x,si1) ?= d(y,si1) = d(y,Ai).
i=1
(Mij∪ Nij), with j ∈ {2,...,φ − 1}.
i=1Ai− ∪ϑ
i=1Bi− ∪φ−1
i=2Ci. To
10

Page 11

Case 2: x,y are extreme vertices. If x,y belong to the same block of
G, then x,y belong to different sets of Π. On the contrary, if x,y belong to
different blocks in G, then let us suppose there exists an extreme vertex c such
that d(x,c) ≤ 1 or d(y,c) ≤ 1. We can suppose c ∈ Ai, for some i ∈ {1,...,ζ},
or c ∈ Bj, for some j ∈ {1,...,ϑ}. Without loss of generality, we suppose
that d(x,c) ≤ 1. Since x and y belong to different blocks of G, we have
d(y,c) > 1. So we obtain either d(x,Ai) = d(x,c) ≤ 1 < d(y,c) = d(y,Ai) or
d(x,Bj) = d(x,c) ≤ 1 < d(y,c) = d(y,Bj).
Now, if there exists no such a vertex c, then there exist two blocks H,K ?∈
Q with x ∈ H and y ∈ K, which contain more than one cut vertex and only
one extreme vertex. So x,y ∈ A. Let u ∈ H be a cut vertex such that
d(y,u) = maxv∈Hd(y,v). Hence, there exists an extreme vertex si1such that
u belongs to a shortest x−si1path and d(y,si1) = d(y,u)+d(u,si1). As x,y
belong to different blocks and d(y,u) = maxv∈Hd(y,v) we have d(y,u) ≥ 2.
Thus,
d(y,Ai) = d(y,si1)
= d(y,u) + d(u,si1)
≥ 2 + d(u,si1)
> 1 + d(u,si1)
= d(x,u) + d(u,si1)
= d(x,Ai).
Hence, we conclude that if φ ≥ 3, then for every x,y ∈ V , r(x|Π) ?= r(y|Π).
Therefore, Π is a resolving partition.
On the other hand, if φ ≤ 2, then Π′= {A,A1,A2,...Aζ,B1,B2,...,Bϑ}
is a partition of V . Proceeding as above we obtain that Π′is a resolving
partition.
The above bound is achieved, for instance, for the graph in Figure 5,
where ζ = 3, ϑ = 0 and φ = 3. Also, notice that for the particular case of
trees we have ζ = ξ, φ = θ and ϑ = 0. So the above result leads to Corollary
2.
References
[1] R. C. Brigham, G. Chartrand, R. D. Dutton, P. Zhang, Resolving dom-
ination in graphs, Mathematica Bohemica 128 (1) (2003) 25–36.
11

Page 12

[2] J. Caceres, C. Hernando, M. Mora, I. M. Pelayo, M. L. Puertas, C.
Seara, D. R. Wood, On the metric dimension of Cartesian product of
graphs, SIAM Journal of Discrete Mathematics 21 (2) (2007) 273–302.
[3] J. Caceres, C. Hernando, M. Mora, I. M. Pelayo, M. L. Puertas, C.
Seara, On the metric dimension of some families of graphs, Electronic
Notes in Discrete Mathematics 22 (2005) 129–133.
[4] G. Chappell, J. Gimbel, C. Hartman, Bounds on the metric and parti-
tion dimensions of a graph, Ars Combinatoria 88 (2008) 349–366.
[5] G. Chartrand, L. Eroh, M. A. Johnson, O. R. Oellermann, Resolvabil-
ity in graphs and the metric dimension of a graph, Discrete Applied
Mathematics 105 (2000) 99–113.
[6] G. Chartrand, C. Poisson, P. Zhang, Resolvability and the upper di-
mension of graphs, Computers and Mathematics with Applications 39
(2000) 19–28.
[7] G. Chartrand, E. Salehi, P. Zhang, The partition dimension of a graph,
Aequationes Mathematicae (1-2) 59 (2000) 45–54.
[8] M. Fehr, S. Gosselin, O. R. Oellermann, The partition dimension of
Cayley digraphs Aequationes Mathematicae 71 (2006) 1–18.
[9] F. Harary, R. A. Melter, On the metric dimension of a graph, Ars Com-
binatoria 2 (1976) 191–195.
[10] T. W. Haynes, M. Henning, J. Howard, Locating and total dominating
sets in trees, Discrete Applied Mathematics 154 (2006) 1293–1300.
[11] B. L. Hulme, A. W. Shiver, P. J. Slater, A Boolean algebraic analysis
of fire protection, Algebraic and Combinatorial Methods in Operations
Research 95 (1984) 215–227.
[12] M. A. Johnson, Structure-activity maps for visualizing the graph vari-
ables arising in drug design, Journal of Biopharmaceutical Statistics 3
(1993) 203–236.
[13] M. A. Johnson, Browsable structure-activity datasets, Advances in
Molecular Similarity (R. Carb´ o–Dorca and P. Mezey, eds.) JAI Press
Connecticut (1998) 153–170.
12

Page 13

[14] S. Khuller, B. Raghavachari, A. Rosenfeld, Landmarks in graphs, Dis-
crete Applied Mathematics 70 (1996) 217–229.
[15] R. A. Melter, I. Tomescu, Metric bases in digital geometry, Computer
Vision Graphics and Image Processing 25 (1984) 113–121.
[16] V. Saenpholphat, P. Zhang, Conditional resolvability in graphs: a sur-
vey, International Journal of Mathematics and Mathematical Sciences
38 (2004) 1997–2017.
[17] P. J. Slater, Leaves of trees, Proc. 6th Southeastern Conference on Com-
binatorics, Graph Theory, and Computing, Congressus Numerantium 14
(1975) 549–559.
[18] I. Tomescu, Discrepancies between metric and partition dimension of a
connected graph, Discrete Mathematics 308 (2008) 5026–5031.
13