1. Introduction
The problem of finding one optimal location for schools, drug stores, police stations, and hospitals requires facilities to be placed near the users in order to minimize, for example, the distance traveled to reach them. Location theory deals with this type of optimization problem. Location functions such as the median, the center, and the mean have been used to solve these type of problems. On the other hand, there are circumstances where placing one or more facilities as far as possible from the users is the best solution. For instance, it is necessary to locate nuclear power plants far from cities or towns to minimize the risk of radiation problems. Similar problems include the determination of suitable locations for observatories, radio stations, airports, and chemical plants. The solution to the problem of finding an optimal location for these types of obnoxious facilities on networks has been studied by Church and Garfinkel [1] , Minieka [2] , Ting [3] , and Zelinka [4] . In these investigations two solutions to the problem are given from an algorithmic perspective. The most appealing solution is called the antimedian, the points that maximizes the total distance from the facility to the users. Another solution is the anticenter, the points that maximizes the total distance from facilities to users. For more information about obnoxious facilities the reader is remitted to [5] -[7] . In the case of tree graphs, Ting [3] published a linear algorithm to find the antimedian of a tree, and Zelinka [4] proved that the set of leaves of a tree contains an antimedian. This problem can be approached through the axiomatization of location functions. The input of a location function consists of some information with respect to the users of the facilities, and the output is related to the consensus reached based on the given information. The rationality of this process is supported by the fact that location functions must satisfy a number of consensus axioms. The mean function on tree graphs was the first location function studied from the axiomatic point of view by Holzman [8] in the continuous case (in the continuous case a tree contains an infinity number of elements, the edges of the tree are considered to be rectifiable curves, and a profile
and its members are allowed to be located anywhere on edges). After that Vohra [9] also characterized the median function in the continuous case; in addition the reader can see [10] . In the discrete case, the center, the median, and the mean function have been characterized axiomatically on trees (see for example [11] -[20] ). Not much research has been done with respect to the axiomatic characterization of obnoxious location functions, but recently Balakrishnan et al. [21] published a characterization of the antimedian function on paths. In what follows, we present a different axiomatic characterization, also on paths, of the antimedian function. Ortega and Wang have recently sent for publication an axiomatic characterization of the antimean function on paths. For more information about location theory and axiomatization we refer the reader to the following references [22] -[27] .
2. Preliminaries
Let
be a finite metric space and set
![](https://www.scirp.org/html/htmlimages\4-1200179x\79544f35-5758-4ea0-a34e-a06f6443c8a6.png)
where
is the cartesian product of
. The elements of
are called
and usually denoted by
. Location theory and consensus theory are related to solve the following problem: Given a collection of
users (voters, customers, clients, etc.) with each user having a preferred location point in
, one attempts to find a set of elements of
that satisfy the preferences of the users with respect to some well-defined criteria. Modeling this situation requires the use of a location function on
, which is a function
, where
denotes the set of all subsets of
. Three well known examples of location functions are:
a) the center function, denoted by Cen, and defined as
![](https://www.scirp.org/html/htmlimages\4-1200179x\46a3d180-2e46-4e1c-acfb-83033bbae483.png)
where
.
b) the median function, denoted by Med, and defined as
![](https://www.scirp.org/html/htmlimages\4-1200179x\2c1e7fb1-5746-4ead-b127-1c45c6321fb3.png)
where
.
c) the mean function, denoted by Mean, and defined as
![](https://www.scirp.org/html/htmlimages\4-1200179x\474c2c32-c38b-48bf-aadd-44049138255d.png)
where
.
We are interested in finite metric spaces defined in terms of connected graphs. Let
be a finite connected graph, and let
be the usual distance on
, where
is the length of a shortest path between
and
. It is well known that
is a metric space, and observe that a profile in a graph
is simply a sequence of vertices where repetitions are allowed. We will investigate some properties of the antimedian function on finite metric spaces defined in terms of a very special type of connected graphs, namely paths.
3. The Antimedian Function on Paths
In this section
or
will denote a path of length
. We will label the vertices of
as
and assume that the order that the vertices have in the path is given by the order of the numbers
. Hence,
will be represented as
(1)
Notice that the set of vertices is
and also that vertex
is adjacent to vertex
, vertex
is adjacent to vertex
and so on. In the case
has an even number of vertices, we will write
. In the case
has an odd number of vertices, we will write
. Let
be a profile on
; for any
we define the status of
with respect to
to be the number
![](https://www.scirp.org/html/htmlimages\4-1200179x\56c724ba-995b-4928-9dea-2aca485a0fea.png)
A vertex
is called an antimedian of
if
![](https://www.scirp.org/html/htmlimages\4-1200179x\3afdd54b-bdbe-4936-a936-d6372bc3f217.png)
The antimedian of
is the set
![](https://www.scirp.org/html/htmlimages\4-1200179x\8f76cae2-53f7-4304-ace1-f06a842b4915.png)
In order to study the antimedian function on
, we will divide the paths in two classes. The set of paths that have an odd number of vertices will be called odd paths, and the set of paths with an even number of vertices will be called even paths. Let
be a profile on
, the notation
![](https://www.scirp.org/html/htmlimages\4-1200179x\ee83e78b-977f-48cc-b5d7-eec5768cb70c.png)
will indicate that there is
such that
. We also use
to denote the set of all the different vertices included in
, and the number of vertices in the profile
counting repetitions is denoted by
.
For example consider the profile
on
with
. In this case
,
, and
.
If
and
are profiles on
, denote by
the profile
![](https://www.scirp.org/html/htmlimages\4-1200179x\8b2fb329-c218-4cf9-abd1-52b0bf26c497.png)
The profile
is called the concatenation of
and
. The following result related to the antimedian function has been proved in [21] .
Lemma 1 Let
and
be profiles on
. If
, then
![](https://www.scirp.org/html/htmlimages\4-1200179x\0771f5d6-300d-496a-a46b-86a9493be789.png)
The definition of the antimedian function implies the following characteristic of this function.
Lemma 2 Let
be a profile on
, and let
be any permutation of
. Then
![](https://www.scirp.org/html/htmlimages\4-1200179x\d9f6f654-eb27-456b-a801-0d9567516037.png)
where ![](https://www.scirp.org/html/htmlimages\4-1200179x\3c0b6cc7-3921-4559-8a14-9bc7525cf0dd.png)
The median function on finite tree graphs satisfies the following property that was proved in [13] , and will be important in the proof of several results.
Lemma 3 Let
be a profile on a finite tree
. If
and if
is a path contained in
such that
, then
![](https://www.scirp.org/html/htmlimages\4-1200179x\dd2e3b35-af53-4722-8afd-ccec403cfea0.png)
The property of the median function described by Lemma 3 will be called the increasing status property.
Lemma 4 Let
be a profile on a path
of length
. Then a)
implies
b)
implies
.
Proof. Notice first that a path is also a tree; consequently, we can apply to
the increasing status property. We first obtain the set
; if
for some
, then we define the paths
![](https://www.scirp.org/html/htmlimages\4-1200179x\5f6bef33-b7dc-4edb-a125-39a6aa31be1d.png)
and
![](https://www.scirp.org/html/htmlimages\4-1200179x\de3687d4-5c15-4649-a87a-fd11de7a3ad8.png)
By the increasing status property we have
![](https://www.scirp.org/html/htmlimages\4-1200179x\f20a7f54-0133-4409-a04b-4158675d1e5c.png)
and
![](https://www.scirp.org/html/htmlimages\4-1200179x\526d7131-cb89-49b1-ae3c-7ffcf1f3a6b4.png)
Observe that if
, then
, and if
, then
.
On the other hand, assume
. Define the paths
![](https://www.scirp.org/html/htmlimages\4-1200179x\43630d5f-dfcd-4aaa-bb82-78469088c642.png)
and
![](https://www.scirp.org/html/htmlimages\4-1200179x\4c80e936-016a-47e9-9aa1-e42e89154795.png)
By the increasing status property we have
![](https://www.scirp.org/html/htmlimages\4-1200179x\689ef261-64d7-4c83-afbb-86f2280c8a2b.png)
and
![](https://www.scirp.org/html/htmlimages\4-1200179x\feca7ba6-a2e1-47c7-aa36-c519ae116b5f.png)
If
, then
, and if
, then
. ![](https://www.scirp.org/html/htmlimages\4-1200179x\5182859e-2f27-4473-880c-2617738c1193.png)
We say that a profile
on
is of the form
for some integer
, if
contains exactly
times the vertices
and
. For example the profile
is of the form
. Profiles
of the form
are special for the antimedian functions because
.
Lemma 5 Let
be a profile the form
for some integer
on a path
of length
. Then
.
Proof. It is well known that if
is a profile on a finite tree
, the median of
consists of all the vertices in the path
![](https://www.scirp.org/html/htmlimages\4-1200179x\623e79db-9898-466c-ba96-9b448b61d991.png)
from
to
. This implies that
![](https://www.scirp.org/html/htmlimages\4-1200179x\b74e0a9c-5b7e-490d-9b2e-1976d9c05436.png)
Since a path
is also a tree, and if
, then we have
![](https://www.scirp.org/html/htmlimages\4-1200179x\7b483403-96bd-46d7-baa3-64412f6f6da3.png)
and the definition of the antimedian function implies that
. Because
is of the form
, then
, and we can reorder the vertices of
to define the profile
![](https://www.scirp.org/html/htmlimages\4-1200179x\af17f7c4-fa76-4f06-95e3-11e991908778.png)
By Lemmas 1 and 2 we obtain
![](https://www.scirp.org/html/htmlimages\4-1200179x\5e82f335-a7b8-4dfb-abcc-77aa05cb3ae4.png)
![](https://www.scirp.org/html/htmlimages\4-1200179x\b7df11d5-a7ef-4cd6-8941-c03e96b9d126.png)
The next result characterizes profiles
on a paths of length
that satisfy the condition
, and that are not of the form
.
Lemma 6 Let
be a path of length
, and let
be a profile that is not of the form
for some integer
. If
, then
.
Proof. Since
is a tree we can apply the increasing status property. We start by obtaining the set
. If
for some
, then we define the paths
![](https://www.scirp.org/html/htmlimages\4-1200179x\9e21bb42-4d8c-46e7-b04b-42f07b53ae2d.png)
and
![](https://www.scirp.org/html/htmlimages\4-1200179x\16604e0e-5835-40aa-b87a-641dd6d0c9d9.png)
By the increasing status property we obtain
![](https://www.scirp.org/html/htmlimages\4-1200179x\af3d3e3b-fde6-4899-b2a0-417b46c3aff8.png)
and
![](https://www.scirp.org/html/htmlimages\4-1200179x\f4c0c67f-b3a9-48f2-9e35-1d303af7a453.png)
Observe that:
if
, then
. On the other hand, assume
. Define the paths
![](https://www.scirp.org/html/htmlimages\4-1200179x\28e04a88-3b71-4fdb-8434-5c144d4f040f.png)
and
![](https://www.scirp.org/html/htmlimages\4-1200179x\18479e67-8778-48aa-9526-2048ef847b16.png)
By the increasing status property we have
![](https://www.scirp.org/html/htmlimages\4-1200179x\9b15494a-377e-4330-aa90-5cb95514f946.png)
and
![](https://www.scirp.org/html/htmlimages\4-1200179x\2602ac6e-5934-4649-b22f-5afa8d6e5b46.png)
Note that
implies
. ![](https://www.scirp.org/html/htmlimages\4-1200179x\c3d82ff5-9379-4b06-9a64-9a1b14706388.png)
From Lemmas 4, 5, and 6 we obtain the following important result that characterizes the output of the antimedian function on paths of length
.
Lemma 7 If
is a profile on a path
of length
, then
![](https://www.scirp.org/html/htmlimages\4-1200179x\1066df9b-567c-4a17-b6a3-8177b838c257.png)
Assume
![](https://www.scirp.org/html/htmlimages\4-1200179x\a313c09b-e376-4fb7-a063-cd199e9885f6.png)
is a path of length
. If
and
, then
. Similarly, if
and
, then
. Denote by
the set of vertices
such that
; similarly we define the set
. Using the sets
![](https://www.scirp.org/html/htmlimages\4-1200179x\d830d199-92bb-4719-b1e5-cb15375a3982.png)
we define a partition of the profile
as follows: denote by
the profile such that
and each vertex in
is included in the profile
as many times it appears in
. In a similar way we define the profile
using the set
. Notice that
can be seen as the concatenation of profiles
and
, in other words
. The following number
(2)
will play an important role in the following sections.
4. The Antimedian Function on Odd Paths
In this section
![](https://www.scirp.org/html/htmlimages\4-1200179x\634f5f0d-c45b-4199-9e59-f6b7762a59b9.png)
represents a path such that
, and note that in this case
. Let
be a profile on
; we will use
to define a new profile that will be denoted
. This profile contains the vertex
repeated
times. In other words we are assuming that
for all
. So,
is the profile
![](https://www.scirp.org/html/htmlimages\4-1200179x\555f95ce-7fe9-41d0-9c96-b68d3120f5e9.png)
We want to establish a relationship between
and
. From the definition of profiles
and
we derive the identities
(3)
and
(4)
Using (3), (4), and the definitions of
and
, we get
![](https://www.scirp.org/html/htmlimages\4-1200179x\3d5cd456-27d2-469f-a2ec-c1088d9d6296.png)
In terms of
, defined by (2), and
we deduce the following relation for ![](https://www.scirp.org/html/htmlimages\4-1200179x\5b20999a-4e91-480c-af4e-e3997d24a01c.png)
![](https://www.scirp.org/html/htmlimages\4-1200179x\b56e4407-d3f8-4332-8c7a-a0409671a1b8.png)
The next result is corollary to the definition of the number
.
Corollary 1 If
is a profile on
, then
if and only if
.
The definition of
and
implies
![](https://www.scirp.org/html/htmlimages\4-1200179x\3ffa585a-46bd-4ad2-a4a7-8e36f19f1882.png)
![](https://www.scirp.org/html/htmlimages\4-1200179x\2ea1f7dd-e402-4c3b-8859-e26838ea0ee5.png)
These relations imply
![](https://www.scirp.org/html/htmlimages\4-1200179x\d1348d64-fd9f-4c0b-89c5-48baf435c5b4.png)
The definition of
and the fact that
indicate
(5)
The following three lemmas establish an important relationship between the numbers
,
, and
. These results will be used to characterize the antimedian of profiles
on
.
Lemma 8 If
is a profile on
, then
if and only if
.
Proof. Assume first
, and notice
![](https://www.scirp.org/html/htmlimages\4-1200179x\ed64e4bd-98ec-4f39-bfc9-1e75c8138f6e.png)
![](https://www.scirp.org/html/htmlimages\4-1200179x\7d91b758-c20b-4594-a566-9d5732f37a06.png)
![](https://www.scirp.org/html/htmlimages\4-1200179x\be61e7be-049d-4012-98a8-d6078c589125.png)
![](https://www.scirp.org/html/htmlimages\4-1200179x\717e8e69-f1ae-414f-ae5e-052a068a0f81.png)
This implies
.
Because of (5) and the fact that
, we obtain
![](https://www.scirp.org/html/htmlimages\4-1200179x\e182e5d2-3af2-40f2-8f37-475ddf35b47d.png)
![](https://www.scirp.org/html/htmlimages\4-1200179x\8d74d8af-bd4a-457f-9fcb-3f13c9392aa0.png)
![](https://www.scirp.org/html/htmlimages\4-1200179x\3b123d89-cb80-428b-9d6a-8f0da433f357.png)
![](https://www.scirp.org/html/htmlimages\4-1200179x\4d224a23-ca9b-47b7-af69-0f873b0c347d.png)
Replacing the equal sign with
and
in the proof of Lemma 8, we obtain the next two results.
Lemma 9 If
is a profile on
, then
if and only if
.
Lemma 10 If
is a profile on
, then
if and only if
.
We end this section with an important result that characterizes the antimedian of a profile
on odd paths that is not of the form
for some integer
.
Lemma 11 Let
be a profile on an odd path
. If
is not of the form
for some integer
, then
![](https://www.scirp.org/html/htmlimages\4-1200179x\4b270b9c-9bb6-4dc7-ae65-d9c295c29e79.png)
Proof. Assuming
and because
is not of the form
, then Lemma 8 implies
, and Lemma 6 proves
.
If
, then Lemma 10 shows
, and Lemma 4 demonstrates
.
If
, then Lemma 9 shows
, and Lemma 4 proves
. ![](https://www.scirp.org/html/htmlimages\4-1200179x\691fe71f-c261-481a-902b-a88241fd7420.png)
5. The Antimedian Function on Even Paths
In this section
![](https://www.scirp.org/html/htmlimages\4-1200179x\4b66cd44-7173-40fa-95d1-9a7f744e0dcb.png)
represents a path where
; so, we have
, and in this case
. Let
be a profile on
. Using similar ideas as in the last section, we can obtain a relationship between the numbers
,
,
, and
. Since the profile
contains all the vertices of
whose index is less or equal to
, then
(6)
Using the profile
we obtain
(7)
From (6) and (7) and the definition of
and
, we deduce
![](https://www.scirp.org/html/htmlimages\4-1200179x\e49e596b-f5e1-4409-9057-7bd15c9cd95e.png)
In terms of
, we have the relation
(8)
Observe that
![](https://www.scirp.org/html/htmlimages\4-1200179x\c8dd4eab-ef01-49e7-b7d8-b2506aaecdef.png)
![](https://www.scirp.org/html/htmlimages\4-1200179x\9fc6cd31-0cfb-4d6d-93b7-808c11624bd3.png)
Using a similar argument as above, we obtain
![](https://www.scirp.org/html/htmlimages\4-1200179x\6ec49d8f-6af5-4be3-b03e-82e4ad69b998.png)
The definition of
implies the identity
![](https://www.scirp.org/html/htmlimages\4-1200179x\687735c4-25b7-4bc0-a15d-7538ee81a6eb.png)
This identity provides the following relation between
and
.
(9)
The next three results show some properties of the numbers
,
,
, and
.
Lemma 12 If
is a profile on
, then
if and only if
.
Proof. Assume first
, and notice that (8) and (9) indicate
![](https://www.scirp.org/html/htmlimages\4-1200179x\15acf6ed-6c38-44ef-acb3-ee30a6527852.png)
![](https://www.scirp.org/html/htmlimages\4-1200179x\df1a05ae-3486-4537-8861-445a5dd7f187.png)
![](https://www.scirp.org/html/htmlimages\4-1200179x\72f1731d-10ae-465f-9cba-e1ceb53ddfac.png)
![](https://www.scirp.org/html/htmlimages\4-1200179x\6bc84ecb-6bbd-4fb8-b1d0-0f31653c0b9a.png)
![](https://www.scirp.org/html/htmlimages\4-1200179x\8925dca7-c008-4649-bd9f-daf31a587922.png)
Conversely, if
, then
![](https://www.scirp.org/html/htmlimages\4-1200179x\b65e6efe-7f9d-4589-8b78-c79ff27a06de.png)
![](https://www.scirp.org/html/htmlimages\4-1200179x\27915fb8-e0c1-44a3-a836-6ea7858c63bb.png)
![](https://www.scirp.org/html/htmlimages\4-1200179x\00cef252-936a-41ef-a019-bde46f2efff3.png)
![](https://www.scirp.org/html/htmlimages\4-1200179x\241fe67f-f93d-439d-ba0d-f250f1c501bf.png)
![](https://www.scirp.org/html/htmlimages\4-1200179x\f1f9c560-004b-46dd-9f92-9a3f43fd2e55.png)
![](https://www.scirp.org/html/htmlimages\4-1200179x\68325060-aca8-41bf-86b0-c3fb78174da3.png)
By replacing the equal sign with
and
in the proof of Lemma 12, we obtain the following two results.
Lemma 13 If
is a profile on
, then
if and only if
.
Lemma 14 If
is a profile on
, then
if and only if
.
The next lemma is an important result because it characterizes the antimedian of profiles
, on even paths, that are not of the form
for some integer
.
Lemma 15 Let
be a profile on an even path
. If
and
is not a profile of the form
for some integer
, then
![](https://www.scirp.org/html/htmlimages\4-1200179x\37260f97-7343-4b67-ba90-9e014f0fbeb9.png)
Proof. Assuming
and since
is not of the form
, Lemma 12 shows
and Lemma 6 indicates
.
If
, then Lemma 14 implies
. Now Lemma 4 proves
.
If
, then Lemma 13 shows
, and Lemma 4 implies
. ![](https://www.scirp.org/html/htmlimages\4-1200179x\685d615d-ac94-41b8-8d91-7d9687c69062.png)
The next result is a corollary to Lemma 15.
Corollary 2 Let
be a profile on
. If
is of the form
for some integer
, then
.
Proof. Notice that in this case the profile
contains
times the vertex
, and the profile
contains
times the vertex
. Consequently, we have
![](https://www.scirp.org/html/htmlimages\4-1200179x\56cff442-c280-40ef-9f89-3d2d352b8970.png)
Finally, Lemma 15 implies
. ![](https://www.scirp.org/html/htmlimages\4-1200179x\f48ba338-c6a3-4c0b-9106-2838c7c96f2d.png)
6. The Axioms and the Main Result
The axioms listed below are among the desirable properties that a general location function should satisfy, and it is not difficult to verify that the antimedian function satisfies these properties.
Oddness (O): Let
be a location function on a path
of length
with
. Let
be defined as in (2); if
is not a profile of the form
for some integer
, then
![](https://www.scirp.org/html/htmlimages\4-1200179x\cc3dc32c-27bb-4c2d-a48a-f4ff4b923426.png)
Evenness (E): Let
be a location function on a path
of length
with
. Let
be defined as in (2); if
is not a profile of the form
for some integer
, then
![](https://www.scirp.org/html/htmlimages\4-1200179x\a57cd4b5-41b3-40be-b3d8-59a9e1b21ca9.png)
Consistency (C): Let
be a location function on
. If
and
are profiles and
with
, then
.
Extremeness (Ex): Let
be a location function, and
be a profile on
. If
, then
.
Generalized Extremeness (G-Ex): Let
be a location function, and let
be a profile on
. If
is of the form
for some integer
, then
.
Anonymity (A): For any profile
on
and any permutation
of
, we have
, where ![](https://www.scirp.org/html/htmlimages\4-1200179x\993a75de-9d11-40a5-8de7-db60629b8045.png)
Some of these axioms are not independent. For example it is clear that (Ex) is a particular case of (G-Ex) when
. Next we prove that if a location function satisfies (C) and (Ex), it also satisfies (G-Ex).
Lemma 16 If
is a location function on
that satisfies axioms (C), (A), and (Ex), then
satisfies axiom (G-Ex).
Proof. Let
be a profile on
of the form
for some integer
. Corollary 2 implies
. Because of axiom (A), we can reorder the vertices of
to obtain a profile
of the form
. We can express
as a concatenation of profiles of the form
; in other words
. Since
satisfies axiom (Ex), then
, and applying axiom (C) we conclude
![](https://www.scirp.org/html/htmlimages\4-1200179x\f3aaef54-03cb-4e5b-8d08-c8ad512f2cd7.png)
![](https://www.scirp.org/html/htmlimages\4-1200179x\fa750b0a-79b0-4345-a9c4-fdcbab9b2a3c.png)
With the axioms listed above we will give two axiomatic characterizations for the antimedian function. The next theorem contains the first of these characterizations.
Theorem 1 Let
be a location function on
.
is the antimedian function on
if and only if
satisfies axioms (O), (E), (Ex), (C), and (A).
Proof. It is clear that if
is the antimedian function, then
satisfies axioms (O), (E), (Ex), (C), and (A). Assume now
is a location function that satisfies axioms (O), (E), (Ex), (C), and (A). To prove that
is the antimedian function we consider three cases.
Case 1. Assume first
is a profile of the form
for some integer
, by Lemma 5 we have
. Since
satisfies axioms (C), (A), and (Ex), then Lemma 16 proves
satisfies (G-Ex) which implies
. It is clear that in this case
.
Case 2. Assume
is a path such that
. Let
be a profile on
that is not of the form
for some integer
. Notice that
, and if
, then Lemma 8 shows
, and Lemma 6 implies
. On the other hand, since
satisfies axiom (O) and
, then
. Therefore,
means
.
If
, then Lemma 10 indicates
, and Lemma 4 proves
. Since
satisfies axiom (O) and
, we obtain
. From this we conclude that if
, then
. A similar argument can be used to show that if
, then
.
Case 3. Assume
is a path such that
which means
, and let
be a profile on
that is not of the form
for some integer
. If
, then Lemma 12 demonstrates
, and Lemma 6 proves
. Since
satisfies axiom (E) and
we get
. Therefore,
implies
.
If
, then Lemma 13 indicates
, and Lemma 4 shows
. Because
satisfies axiom (E) and
,
. Hence, when
is a profile that is not of the form
and
,
. A similar argument can be used to show that if
, then
. Notice that Cases 1, 2, and 3 demonstrate the theorem. ![](https://www.scirp.org/html/htmlimages\4-1200179x\9f5f500c-fbbc-42f0-a9fc-5d23186d83c5.png)
We leave it to the reader to prove that the axioms used in the proof of Theorem 1 are independent. Notice that in the proof of Theorem 1 we needed to use three axioms to establish Case 1. We want to improve the demonstration of this result using fewer axioms. We achieve this objective using axiom (G-Ex) in the following theorem which is our main result.
Theorem 2 Let
be a location function on
.
is the antimedian function on
if and only if
satisfies axioms (O), (E), and (G-Ex).
Proof. It is clear that if
is the antimedian function, then
satisfies axioms (O), (E), and (G-Ex). Assume now that
is a location function that satisfies axioms (O), (E), and (G-Ex). To prove that
is the antimedian function we consider three cases.
Case 1. Assume first
is a profile of the form
, then by Lemma 5 we obtain
. Because
satisfies axiom (G-Ex), we have
. So in this case
.
Case 2. Assume
is a path such that
which means
. Let
be a profile on
that is not of the form
. If
, then Lemma 8 indicates
, and Lemma 6 implies
. Since
satisfies axiom (O) and
,
. Therefore,
indicates
. A similar argument can be employed to show that if
, then
, and if
, then
.
Case 3. Assume
is a path such that
, and let
be a profile on
. Notice that
.
If
, then Lemmas 12 implies
, and Lemma 6 shows
. Because
satisfies axiom (E), we conclude
. Therefore,
implies
.
A similarly argument can be used to demonstrate that if
, then
, and if
, then
. It is clear that Cases 1, 2, and 3 finish the proof of the theorem. ![](https://www.scirp.org/html/htmlimages\4-1200179x\d6465ad0-e551-4fc5-9925-7e82e75a2be5.png)
Notice that the definition of axioms (O), (E), and (G-Ex) indicate that they are independent. So it is not necessary to add a proof for the independence of these three axioms. More research is needed to find an axiomatic characterization of the antimedian function on tree graphs.