International Journal Series in Multidisciplinary Research (IJSMR), ISSN: 2455–2461
Vol. 1, No. 3, 2015, 17-30 http://ijseries.com/
Graph Theory in Distribution and Transportation Problems and the
Connection to Distance-Balanced Graphs
Hassan Kharazi12, Ehsan Pourhadi*12
1
School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16846-13114, Iran
2
Department of Mathematics and Statistics, Imam Hossein University, Tehran, Iran
ABSTRACT: One of the important issues in everyday life is optimization problem and reduction
of the cost of distribution and transportation of goods. Researchers are always in line with this
objective, by providing search tools trying various approaches to minimize such costs. The
purpose of this paper is to examine the problem and its solution to the modeling induced by graph
theory, algorithms and theorems related. Also, using the properties of distance-balanced graphs we
show that developing the distance-balanced structures in matters of transport networks and similar
issues can be considered as a new way to reduce the certain costs.
MATHEMATICS SUBJECT CLASSIFICATION (2010): 05C12, 94C15, 05C85.
KEYWORDS: Graph theory, distance-balanced graphs, optimization problem, transportation,
Dijkstra and Kruskal algorithms.
1. INTRODUCTION
During the last decades, graph theory has attracted the attention of many researchers.
Graph theory has provided a very nice atmosphere for research of provable techniques in
discrete mathematics for researchers. Moreover, many applications in the computing,
industrial, natural and social sciences are studied by graph theory. It is worth mentioning
that all graphs are usually classified when we encounter to special graphs in modeling of
phenomena in real life. Recently, a new category of graphs, so-called distance-balanced
graphs, has been presented and their properties have been examined in various studies (see
the papers in reference).
In what follows, we introduce this type of graphs and some important results of the
properties of this class of graphs which will be required for our analysis in the future. In
the second section, we focus on the optimization of network structures such as large-scale
distribution networks using the distance-balanced properties. In the last section, the
optimization problem and reduce the costs in the transport using Kruskal and Dijkstra's
*
Corresponding author, Ph.D.,
E-mail: [email protected]
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International Journal Series in Multidisciplinary Research (IJSMR), ISSN: 2455–2461
Vol. 1, No. 3, 2015, 17-30 http://ijseries.com/
algorithms will be investigated. Also, the connection between optimization in
transportation and distance-balanced infrastructures are described and some results are
concluded.
For any edge
, that is:
of graph
let
be the set of vertices which are closer to
than
.
where
Also, let
is the distance of
and
. Similarly,
can be defined:
be the vertices which their distance to both
and
are the same:
We remark that the mentioned sets reveal a partition for the arbitrary connected graph
and also play an important role in theory of metric graphs. For the convenience, the
notation
can be ignored and the sets
,
,
can be applied.
Definition 1.1. A connected graph
is called distance-balanced (DB for short) if and
only if for each edge
of
we have
W ab  W ba .
Handa (1999), initially was the first one who considered distance-balanced partial
cubes and proved that they are 3-connected. One way to identify this category of graphs is
utilizing the total distance Graph and based on this concept, such graphs can be
understood.
We recall that for simple and connected graph , if
then total distance DG(u) is
defined by
Theorem 1.2 (Balakrishnan 2009). Let G be a connected graph. Then G is distancebalanced if and only if
.
In other word, the distance-balanced graphs are precisely the graphs in which all the
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vertices have the same total distance. Another efficient indicator to identify distancebalanced graphs is Wiener index which is defined as the following form.
Let
be a graph on
vertices, and let V1, V2 ⊂ V(G) be -sets of vertices of
such that V1 ∪ V2 = V(G) (note that this implies V1 ∩ V2 = ∅). Then we say that {V1, V2}
is a half-partition of . The opportunity index of a graph
is defined as
We recall that relative Wiener index of
is given by
where
is the set of all 2-element subsets of . We always can obtain graphs
with
opp(G) arbitrarily large. For instance, if
is the corona on
(the graph obtained from
the complete graph on n vertices by attaching a leaf to each vertex), then opp(G) = n(n −
1). We conclude the section with the following key definition: a graph
(of even order)
is an equal opportunity graph if opp (G) = 0.
As we explain it later, the opportunity index of graph can be attractive for some
reasons; loosely speaking, in the definition of this concept, more difference in the term
leads to a graph with lower symmetric metric. When a graph is a model for a real-life
problem (say in economy, location theory, or social choice phenomena) then the network
opportunity measures the unfairness or social inequality of a given topology. In other word,
when we face to real-life problem with apparently immeasurable variables, this tool can be
a good gauge for measuring these variables. Thus, in many situations the design of equal
opportunity networks is highly desirable. According to the above motivation we now
define a graph
(with even order) is a graph of opportunity if opp(G) = 0. As we said
before all distance-balanced graph can be identified by opportunity index. Here, we give
this characteristic result.
Theorem 1.3 (Balakrishnan 2014). A graph
is an equal opportunity graph if and only if
is a distance-balanced graph of even order.
Now, we focus on one of the major problems in graph theory which we call the Wiener
game and this problem has an important relationship with the distance-balanced graphs.
This game is played on a connected graph
of even order. Vertices are chosen, one at
a time, by two players player
and player . Player
starts the game and the players
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International Journal Series in Multidisciplinary Research (IJSMR), ISSN: 2455–2461
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alternate by taking turns choosing a vertex from
until all the vertices have been
selected. Let VA and VB be the sets of vertices selected by players A and B, respectively.
Since the order of G is even, |VA| = |VB|. The goal of both players is to make
as small as possible, respectively. Assuming that both were playing optimally and that sets
VA and VB were selected by the two players, we discuss on the sets
and
.
We say that player A (resp. B) wins the game if WA(G) < WB(G) (resp. WB(G) < WA(G)),
otherwise the game is a draw. Notice that |WA(G)−WB(G)| ≤ opp(G), and in practical
situations, the player who wins the game, often wants to maximize |WA(G)−WB(G)|. It is
worth mentioning that in this game, finding optimum strategies for each player in special
classes of graphs is challenging. The following observation is a direct consequence of
Theorem 1.3.
Corollary 1.4 (Balakrishnan 2014). If
is a distance-balanced graph of even order, then
the Wiener game on
is a draw, regardless of the strategy used by either of the players.
2. APPLICATION OF DISTANCE-BALANCED GRAPHS IN COMPUTAIONAL
OPTIMIZATION FOR MEASURING OF DESIGNED LARGE-SCALE NETWORKS
As observed in previous section, total distance in a network is given by
To compare two different design of a network, total distance as a tool, is an important
factor. Another measure that can be taken into account cost-efficiency ratio (CER) for the
two networks, which are defined as follows:
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So, this measure can be obtained by computing the total
distance of network and related costs. Performance indicator
in real-life problem is relevant to determine the most costeffective strategy for a given budget. According to the results
for distance-balanced graphs, the all total distance of vertices
are the same. This property implies a computational
optimization in order to attain an optimal cost-efficiency ratio.
For instance, consider a network with twenty nodes with
certain links as shown in the figure. We easily see that total
distance of each arbitrary vertex
of the network is equal
to:
Figure 1 node u and d(u,v) as
label of all nodes v
So, assuming a constant cost for any created link between two nodes ( ), this index equals
to:
Therefore, by taking an initial budget (α) to construct a hypothetical link between two
nodes via optimization, if the number of nodes in the distance-balanced graph on a large
scale is , then:
where
is an arbitrary vertex, and this means that for the calculation of such ratio we
only need to calculate the total distance of vertices arbitrary vertex . This represents a
very significant reduction in the computational cost of distance-balanced structures. In the
following example we mention another aspect of properties of distance-balanced
structures:
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International Journal Series in Multidisciplinary Research (IJSMR), ISSN: 2455–2461
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Example 2.1. Suppose two companies A and B
distribute their electronic products to the buyers
including all the sellers of electronics in Tehran. In order
to maintain a balance in the supply of products between
all sales terminals and fair service to citizens, the two
companies intend to fairly distribute its products
between such terminals. The purpose of each of the two
companies is minimization of losses and damages caused
during shipping lines. If sales terminals and transmission
lines are assumed as vertices and edges of a graph,
respectively, then one should minimize the total distance
of vertices of graphs with respect to aim of both
companies. On the other hand, the intention is that no
Figure 2. A half-partition for two
company has incurred losses in this context. The
companies A and B
distance-balanced structures satisfy this goal. Since, any
choice in these structures (including half of the
mentioned terminals by each of companies) induces same and fair damages (see Theorem
1.2 and Corollary 1.4) and the total losses in the calculation of distances between
transmission points become fairly divided between the two companies. See the following
numerical example for a given graph.
Suppose
and
of sale terminals that companies A and B respectively to the
terminals with intent to distribute their products. Obviously, given the assumption of
fairness supply of products we must have the sets
and
with the same cardinal.
Hence, for a distribution to like what we have seen in the figure as above we obtain that
and this verifies our claim that any selection of sales terminals in Tehran by the
distribution companies implies same and fair damages for the both companies. So, the two
companies in the competition with each other to minimize the damages will always draw
(see Corollary 1.4). Therefore, in order to suffer the same and fair damages, it suffices to
minimize
(or similarly,
). On the other hand, we recall that the median vertex of a
given graph is a kind of vertex with the minimum total distance and so our purpose (or the
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companies) can be turned into finding a median graph including all such vertices.
3. FINDING OPTIMAL PATHS FOR THE TRANSPORTATION PROBLEM AND ITS
CONNECTION TO DISTANCE-BALANCED GRAPHS
Theory of Graphs are used as device for modeling and description of real world
network systems such transport, water, electricity, internet, work operations schemes in the
process of production, construction, etc.
Although the content of these schemes differ
among themselves, but they have also
common features and reflect certain items that
are in the relation between each other. So in
the scheme of transport network might be
considered manufacturing centers, and roads
and rail links connected directly to those
centers. The following example is designed
for the solution of a practical problem to find
a Minimum Spanning Tree by using Kruskal
algorithm and graph search Dijkstra’s
algorithm to find the shortest path between
two points, also, for this case it is developed a
network model of the transportation problem
which is analyzed in detail to minimize Figure 3. A weighted graph as model for analysis
shipment costs.
As we know that graph theory provides many useful applications in operations research,
a graph is defined as a finite number of points (known as nodes or vertices) connected by
lines (known as edges or arcs). In the following for a given graph shown in Fig. 3 we find
a minimum cost to reach the shortest path between two points. There are different path
options to reach from node A to node B, but our goal is to find the shortest path with a
minimum transportation costs, this needs a lot efforts.
3.1 Finding the minimum spanning tree by applying Kruskal algorithm
In what follows the problem of finding the minimum spanning tree for the given case is
described by several figures given in the following. Firstly, we consider all nodes of the
given graph without edges, then we will start to put the edges back in their places starting
from the lowest cost (edge length 1) to the one with higher costs (Figure 4), but notice that
since the technique is optimal we are not allowed to create cycles. This process continues
by placing the second edge of length 2. Edge of lower cost that comes after the ones with
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units 1 and 2 is the edge of length 3. Again we have processed in the same way having in
mind that we must not create cycles.
Figure 4
Figure 5
Figure 6
Applying this rule to all edges of the given Graph given, we have gained a minimum
spanning tree which is given in Figure 7. Edges which are omitted from the graph are
marked by red color, and this happened because their deployment creates cycles Figure 7.
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Figure 7
Figure 8
3.2. Finding the minimum cost path
From the Minimum Spanning Tree shown in Figure 6 we can find the minimum cost
path (trajectory) from node A to node B. As we can see from the Figure 8, there are two
alternative ways to reach from node A to node B ,which can be distinguished by dash
line.
Let’s start with first option to calculate the distance from node A to node B (dash line), the
result is as follows:
µ = 2+3+6+7+4+5+2+5+4+3+6+17 = 64 units
which is the most expensive path . For the second option (bold line):
µ = 3+1+11+7+2 = 24 units
This means that the second option represents the minimum cost path from node A to node
B.
3.3. Application of Dijkstra’s Algorithm
By applying Dijkstra’s Algorithm we are able to find the shortest distances (as the
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length of edge) from a node to all other nodes. Firstly, we start from the node A, which is
chosen as permanent node. By analyzing the distances of the neighborhoods nodes of the
node A, we are able to find the shortest path to node 2 (its distance is equal with 2).
Afterwards node 2 is chosen as permanent node, and we have to check after the distances
from node 2 to the neighbor nodes. To the each neighbor node is added the length of the
permanent node.
Figure 9
Figure 10
Now, since the label of each step is equal to the minimum distance from node A, Figure
10 shows that the minimum distance is chosen as permanent node, and since the 3+2
distance is shorter than 7, this means that distance 7 is not going to be considered anymore
and we have to use the distance 5.
Next, the fixed node is the node with minimum distance 4 from node A, this means that
among all neighbors the node with minimum distance 4 from node A is selected as the
permanent node (Figure 11).
This process is repeated for each node respectively.
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Figure 11
Figure 12
Figure 13
Figure 14
Figure 15
Figure 16
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The Figure 17 is the minimum cost path from
node A to node B. As the conclusion of this
section, we observe that Dijkstra's Algorithm will
find the shortest path between two nodes/vertices.
Kruskal's algorithm will find the minimum
spanning tree connecting all the given vertices.
Basically, Dijkstra's will find a connection
between two vertices, while Kruskal's will find a
connection between and number of vertices. The
results which are obtained for the given example
shows that Dijkstra’s Algorithm is very effective
tool to find the path with lowest cost from node A
to node B. Same results have been obtained also
Figure 17
for Minimum Spanning Tree by using Kruskal
algorithm, but this case the procedure is much
simpler with a minimum spanning tree to reach node B from node A with the lowest total
cost.
Remark 3.1. We note that in the example of this section the weight of each edge is
considered as the cost of transport and similarly, this weight can be considered as a length
of a path, capacity of package of energy, the needed energy for motion and etc.
4. THE PROBLEM OF TRANSPORTATION AND THE RELATION TO DISTANCEBALANCED PROPERTY
Similar to the recent example, the transport problem can be implemented on the
following vertex-transitive weighted graph which is called “truncated tetrahedron”.
In this graph, the distance between any two nodes is
considered based on a cost unit. For example, the
distance between two vertices A and B is 3 units. Due
to the fact that the balance property makes sense in
modeling when the graph is weighted and the concepts
such as distribution and transportation are important,
so two post offices A and B responsible for delivery
are considered as two vertices on this graph.
Given that the purpose of the transport, postal
delivery is at the lowest cost and in view of the fact
that in the city there are two post offices, the cost
should be minimized for each office.
Figure 18
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On the other hand, the balance in transportation
within the city is also desire. Therefore, these offices
should be structured such that both suffered from the
same damages based on Wiener game from Section
1. In order to cover this purpose both offices must
select 6 places (vertices) from 12 places (including
A and B) for delivery. Since the model is distancebalanced so the related results show that any
selection implies the same financial damages for
both post offices. Hence, the location of building is
not important for A and B. For instance, For A and B
as given in Figure 18 if we pick
and
then we obtain the
same cost for both offices:
Figure 19
It is also worth mentioning that
That is, if each office is responsible for delivery of all places then the total imposed cost
will be the same.
Remark 4.1. In a distance-balanced structure, if two post offices are responsible for
delivery of packages to the same number of places, that is, the total number of places is
even, then the postal shipping cost will be the same for both offices and moreover, this is
independent from their locations.
Remark 4.2. Such as what was shown in the recent calculation, either A or B, regardless
of the name of office, the total distance is constant and their locations are not significant.
This fact shows that in the distance-balanced structures, choosing the location for a
particular department such as emergency unit, fire station is not important and this leads to
save the time and cost for construction of the site selection.
REFERENCES
Balakrishnan K., Brešar B., Changat M., Klavžar S., Vesel A., Žigert Pleteršek P. (2014), “Equal opportunity
networks, distance-balanced graphs, and Wiener game”, Discrete Optim. 12, 150–154.
Balakrishnan K., Changat M., Peterin I., Spacapan S., Sparle P., Subhamathi A.R. (2009), “Strongly
distance-balanced graphs and graph products”, European J. Combin. 30, 1048-1053.
29
International Journal Series in Multidisciplinary Research (IJSMR), ISSN: 2455–2461
Vol. 1, No. 3, 2015, 17-30 http://ijseries.com/
Cabello S., Luksic. P. (2011), “The complexity of obtaining a distance-balanced graph”, Electron. J.
Combin. 18 (1), paper 49.
Fukuda K., Handa K. (1993), “Antipodal graphs and oriented matroids”, Discrete Math. 111, 245-256.
Gallian J.A. (2007), “Dynamic Survey DS6: Graph Labeling”, Electronic J. Combin. DS6, 1-58.
Handa K. (1999), “Bipartite graphs with balanced (a,b)-partitions”, Ars Combin. 51, 113-119.
Ilic A., Klavzar S., Milanovic M. (2010), “On distance-balanced graphs”, European J. Combin. 31, 733-737.
Jerebic J., Klavzar S., Rall D.F. (2008), “Distance-balanced graphs”, Ann. Combin. 12 (1), 71-79.
Khalifeh M.H.,Youse-Azari H., Ashrafi A.R., Wagner S.G. (2009), “Some new results on distance-based
graph invariants”, European J. Combin. 30, 1149-1163.
Miklavic S., Sparl P. (2012), “On the connectivity of bipartite distance-balanced graphs”, European J.
Combin. 33, 237-247.
Tavakoli M., Yousefi-Azari H., Ashrafi A.R. (2012), “Note on edge distance-balanced graphs”, Trans.
Combin. 1(1), pp. 1-6.
Yang R., Hou X., Li N., Zhong W. (2009), “A note on the distance-balanced property of generalized Petersen
graphs”, Electron. J. Combin. 16 (1), Note 33.
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