# The Forwarding Indices

of Graphs – a Survey
^{†}^{†}thanks: The work was
supported partially by NNSF of China (No. 11071233).

###### Abstract

A routing of a given connected graph of order is a collection of simple paths connecting every ordered pair of vertices of . The vertex-forwarding index of with respect to is defined as the maximum number of paths in passing through any vertex of . The vertex-forwarding index of is defined as the minimum over all routing ’s of . Similarly, the edge-forwarding index of with respect to is the maximum number of paths in passing through any edge of . The edge-forwarding index of is the minimum over all routing ’s of . The vertex-forwarding index or the edge-forwarding index corresponds to the maximum load of the graph. Therefore, it is important to find routings minimizing these indices and thus has received much research attention in the past ten years and more. In this paper we survey some known results on these forwarding indices, further research problems and several conjectures.

Keywords: Vertex-forwarding index, Edge-forwarding index, Routing

AMS Subject Classification: 05C40

## 1 Introduction

In a communication network, the message delivery system must find a route along which to send each message from its source to its destination. The time required to send a message along the fixed route is approximately dominated by the message processing time at either end-vertex, intermediate vertices on the fixed route relay messages without doing any extensive processing. Metaphorically speaking, the intermediate vertices pass on the message without having to open its envelope. Thus, to a first approximation, the time required to send a message along a fixed route is independent of the length of the route. Such a simple forwarding function can be built into fast special-purpose hardware, yielding the desired high overall network performance.

For a fully connected network, this issue is trivial since every pair of processors has direct communication in such a network. However, in general, it is not this situation. The network designer must specify a set of routes for each pair of vertices in advance, indicating a fixed route which carries the data transmitted from the message source to the destination . Such a choice of routes is called a routing.

We follow [34] for graph-theoretical terminology and notation not defined here. A graph always means a simple and connected graph, where is the vertex-set and is the edge-set of . It is well known that the underlying topology of a communication network can be modelled by a connected graph , where is the set of processors and is the set of communication links in the network.

Let be a connected graph of order . A routing in is a set of fixed paths for all ordered pairs of vertices of . The path specified by carries the data transmitted from the source to the destination . A routing in is said to be minimal, denoted by , if each of the paths specified by is shortest; is said to be symmetric or bidirectional, if for all vertices and , path is the reverse of the path specified by ; is said to be consistent, if for any two vertices and , and for each vertex belonging to the path specified by , the path is the concatenation of the paths and .

It is possible that the fixed paths specified by a given routing going through some vertex are too many, which means that the routing loads the vertex too much. Load of any vertex is limited by capacity of the vertex, for otherwise it would affect efficiency of transmission, even result in malfunction of the network.

It seems quite natural that a “good” routing should not load any vertex too much, in the sense that not too many paths specified by the routing should go through it. In order to measure the load of a vertex, Chung, Coffiman, Reiman and Simon [7] proposed the notion of the forwarding index.

Let be a graph with a give routing and be a vertex of . The load of with respect to , denoted by , is defined as the number of the paths specified by going through . The parameter

is called the forwarding index of , and the parameter

is called the forwarding index of .

Similar problems are studied for edges by Heydemann, Meyer and Sotteau [17]. The load of an edge with respect to , denoted by , is defined as the number of the paths specified by which go through it. The edge-forwarding index of , denoted by , is the maximum number of paths specified by going through any edge of , i.e.,

and the edge-forwarding index of is defined as

Clearly, and . The equality however does not always holds.

The original research of the forwarding indices is motivated by the problem of maximizing network capacity [7]. Maximizing network capacity clearly reduces to minimizing vertex-forwarding index or edge-forwarding index of a routing. Thus, whether or not the network capacity could be fully used will depend on the choice of a routing. Beyond a doubt, a “good” routing should have a small vertex-forwarding index and edge-forwarding index. Thus it becomes very significant, theoretically and practically, to compute the vertex-forwarding index and the edge-forwarding index of a given graph and has received much attention the recent ten years and more.

Generally, computing the forwarding index of a graph is very difficult. In this paper, we survey some known results on these forwarding indices, further research problems, several conjectures, difficulty and relations to other topics in graph theory.

Since forwarding indices were first defined for a graph, that is, an undirected graph [7], most of the results in the literature are given for graphs instead of digraphs, but they can be easily extended to digraphs. Nevertheless, we give here most of the results for graphs as they appear in the literature.

## 2 Basic Problems and Results

2.1. NP-completeness

Chung, Coffiman, Reiman and Simon [7] asked whether the problem of computing the forwarding index of a graph is an NP-complete problem. Following [11], we state this problem as follows.

Problem 2.1 Forwarding Index Problem,

Instance: A graph and an integer .

Question: ?

Heydemann, Meyer, Sotteau and Opatrný [20] first showed that Problem 2.1 is NP-complete for graphs of diameter at least when the routings considered are restricted shortest, consistent and symmetric; a P-problem for graphs of diameter when the routings considered are restricted to be shortest. Saad [27] proved that Problem 2.1 is NP-complete for for general routings even if the diameter of the graph is . However, Problem 2.1 has not yet been solved for graphs of when the routings considered are restricted to be shortest be minimal and/or, consistent and/or symmetric.

The same problem was also suggested by Heydemann, Meyer and Sotteau [17].

Problem 2.2 Edge-Forwarding Index Problem,

Instance: A graph and an integer .

Question: ?

Heydemann, Meyer, Sotteau and Opatrný [20] showed that Problem 2.2 is NP-complete for graphs of diameter at least when the routings considered are restricted to be minimal, consistent and symmetric; a P-problem for graphs of diameter when the routings considered are restricted to be minimal.

2.2. Basic Bounds and Relations

For a given connected graph of order , set

and

The following bounds of and were first established by Chung, Coffiman, Reiman and Simon [7] and Heydemann, Meyer and Sotteau [17], respectively.

Theorem 2.3 (Chung et al [7]) Let be a connected graph of order . Then

(1) |

and the equality is true if and only if there exists a minimal routing in which induces the same load on every vertex. The graph that attains this upper bound is a star .

Theorem 2.4 (Heydemann et al [17]) Let be a connected graph of order . Then

(2) |

and the equality is true if and only if there exists a minimal routing in which induces the same load on every edge. The graph that attains this upper bound is a complete bipartite graph .

Recently, Xu et al. [40] have showed the star is a unique graph that attains the upper bound in (1).

Problem 2.5 Note that the upper bound given in (2) can be attained. Give a characterization of graphs whose vertex- or edge-forwarding indices attain the upper bound in (2).

Although the two concepts of vertex- and edge-forwarding index are similar, no interesting relationships is known between them except the following trivial inequalities.

Theorem 2.6 (Heydemann et al [17]) For any connected undirected graph of order , maximum degree , minimum degree ,

(a) ;

(b) ;

(c) .

The inequality in (a) is also valid for minimal routings.

No nontrivial graph is found for which the forwarding indices hold one of the above equalities. Thus, it is necessary to further investigate the relationships between and or between and .

Problem 2.7 For a graph and its line graph , investigate the relationships between and or between and .

2.3. Optimal Graphs

A graph is said to be vertex-optimal if , and edge-optimal if . Note that if is a routing of such that , then

(3) |

Heydemann et al [17] showed that the equality (3) is valid for any Cayley graph.

Theorem 2.8 Let be a connected Cayley graph with order . Then

(4) |

From Theorem 2.8, Cayley graphs are vertex-optimal. Some results and problems on the forwarding indices of vertex-transitive or Cayley graphs, an excellent survey on this subject has been given by Heydemann [15].

Heydemann et al [20] have constructed a class of graphs for which the vertex-forwarding index is not given by a minimal consistent routing. Thus, they suggested the following problems worthy of being considered.

Problem 2.9 (Heydemann et al [20]) For which graph or digraph does there exist a minimal consistent routing such that or a consistent routing such that ?

Heydemann et al [20] have ever conjectured that in any vertex-transitive graph , there exists a minimal routing in which the equality (4) holds.

The conjecture has attracted many researchers for ten years and more without a complete success until 2002. Shim, Širáň and Žerovnik [28] disproved this conjecture by constructing an infinite family of counterexamples, that is, for any (mod ), where is the generalized Petersen graph and the symbol denotes the strong product.

Gauyacq [12, 13, 14] defined a class of quasi-Cayley graphs, a new class of vertex-transitive graphs, which contain Cayley graphs, and are vertex-optimal. Solé [30] constructed an infinite family of graphs, the so-called orbital regular graphs, which are edge-optimal. We state the results of Gauyacq and Solé as the following theorem.

Theorem 2.10 Any quasi-Cayley graph is vertex-optimal, and any orbital regular graph is edge-optimal.

However, we have not yet known whether a quasi-Cayley graph is edge-optimal and not known whether an orbital regular graph is vertex-optimal. Thus, we suggest to investigate the following problem.

Problem 2.11 Investigate whether a quasi-Cayley graph is edge-optimal and an orbital regular graph is vertex-optimal.

Considering for , Heydemann et al [17] found that the equality (4) is not valid for , and proposed the following conjecture.

Conjecture 2.12 (Heydemann et al [17]) For any distance-transitive graph , there exists a minimal routing for which,

Conjecture 2.13 (Heydemann et al [17]) For any distance-transitive graph , there exists a minimal routing in which we have both

(a) the load of all vertices is the same, and then,

(b) the load of all the edges is almost the same (difference of at most one) and then,

2.4. For Cartesian Product Graphs

The cartesian product can preserve many desirable properties of the factor graphs. A number of important graph-theoretic parameters, such as degree, diameter and connectivity, can be easily calculated from the factor graphs. In particular, the cartesian product of vertex-transitive (resp. Cayley) graphs is still a vertex-transitive (resp. Cayley) graph (see Section 2.3 in [33]). Since quasi-Cayley graphs are vertex-transitive, the cartesian product of quasi-Cayley graphs is still a quasi-Cayley graph. Thus, determining the forwarding indices of the cartesian product graphs is of interest. Heydemann et al [17] obtained the following results first.

Theorem 2.14 Let and be two connected graphs with order and , respectively. Then

(a) ;

(b)

The inequalities are also valid for minimal routings. Moreover, the equality in (a) holds if both and are Cayley digraphs.

Recently, Xu et al [35] have considered the cartesian product of graphs and obtained the following results.

Theorem 2.15 Let . Then

(a) is vertex-optimal and if is vertex-optimal for every then

(b) is edge-optimal and if is edge-optimal for every then

By Theorem 2.15 and Theorem 2.10, the cartesian product of quasi-Cayley graphs is vertex-optimal and the cartesian product of orbital regular graphs is edge-optimal.

## 3 Connectivity Constraint

In this section, we survey the known results of the forwarding indices of -connected or -edge-connected graphs.

3.1. -connected Graphs

Theorem 3.1 If is a -connected graph of order , then

(a) , this bound is best possible in view of (Heydemann et al [17]);

(b) for and diameter , this bound is best possible since it is reached for a wheel of order minus one edge with both ends of degree (Heydemann et al [17]);

(c) for (Heydemann et al [18]);

(d) and this bound is best possible in view of the cycle (Heydemann et al [18]).

Theorem 3.2 (Heydemann et al [18]) for any -connected graph of order with and .

Heydemann et al [18] proposed the following research problem.

Problem 3.3 Find the best upper bound and such that for any -connected graph of order with , and for large enough compared to .

Theorem 3.4 (de la Vega and Manoussakis [9]) For any integer ,

(a) ;

(b) is substantially larger than ; if

(c) .

Recently, Zhou et al. [42] have improved the upper bounds of and in Theorem 3.4 as follows.

Theorem 3.5 If is a -connected graph of order with the maximum degree , then and .

Conjecture 3.6 (de la Vega and Manoussakis [9]) For any positive integer ,

(a) for , which would be best possible in view of the complete bipartite graph ;

(b) there exists a function such that if , then

(c) for , which would be best possible in view of the graph obtained from two complete graphs plus a matching between them, .

It can be easily verified that the conjecture (a) and (c) are true for and . Recently, Zhou et al. [42] have proved that Conjecture 3.6 (a) is true for , that is,

Theorem 3.7 If is a -regular and -connected graph of order . Then .

3.2. -edge-connected Graphs

Theorem 3.8 If is a -edge-connected graph of order , then

(a)

(b) (Cai [3]).

Heydemann et al [17] conjectured that for any -edge-connected graph of order ,

Conjecture 3.9 for any -edge-connected graph of order with and ,

The same problem as the ones in Problem 3.3 can be considered for -edge-connected graphs.

Problem 3.10 Find the best upper bound and such that for any -connected graph of order with , and for large enough compared to .

The following theorem is the only result we have known as far on this problem.

Theorem 3.11 (de la Vega and Manoussakis [9]) For any integer ,

3.3. Strongly Connected Digraphs

It is clear that the notion of the forwarding indices can be similarly defined for digraphs. Many general results, such as Theorem 2.3 and Theorem 2.4 are valid for digraphs. Manoussakis and Tuza [26] consider the forwarding index of strongly -connected digraphs and obtained the following result similar to Theorem 2.4.

Theorem 3.12 Let be a strong digraph of order . Then

(1) , and

(2) The equalities are true if and only if there exists a minimal routing in which induces the same load on every edge.

In addition to validity of Theorem 3.2 for digraphs, they obtained the following results.

Theorem 3.13 Let be a -connected digraph of order , and . Then

(a) ;

(b) for ;

(c) for .

## 4 Degree Constraint

Although Saad [27] showed that for any graph determining the forwarding index problem is NP-complete, yet many authors are interested in the forwarding indices of a graph. Specially, it is still of interest to determine the exact value of the forwarding index with some graph-theoretical parameters. For example, Chung et al [7], Bouabdallah and Sotteau [2] proposed to determine the minimum forwarding indices of -graphs that has order and maximum degree at most . Given and , let

4.1. Problems and Trivial Cases

Problem 4.1 (Chung et al [7]) Given and , determine , and exhibit an -graph and of for which .

Problem 4.2 (Bouabdallah and Sotteau [2]) Given and , determine , and exhibit an -graph and of for which .

For , we can fully connect a graph, i.e., is a complete graph. In this case any routing can be composed only of single-edge paths so that the minimum and is achieved, that is, and for .

For the only connected graph fully utilizing the degree constraint is easily seen to be a cycle. Because of the simplicity of cycles, the for vertex-forwarding index problem can be solve completely for .

Theorem 4.3 For all ,

(a) ;

(b) .

4.2. Results on

Problem 4.1 was solved for or any and with by Heydemann et al [16].

Theorem 4.4 (Heydemann et al [16])

(a) if is even or odd and even, for or for or and ;

(b) if and are odd, for or for and .

Problem 4.1 has not been completely solved for .

Theorem 4.5 (Heydemann et al [16]) For any and ,

(a) for any and such that ;

(b) for any odd such that ;

(c) for any and such that and ;

(d) ;

(e) every -graph such that is -regular and diameter ;

(f) if and are odd.

An asymptotic result on has been given by Chung et al [7].

Theorem 4.6 For any given ,

where the upper bound holds for .

4.3. Results on

Similar to Theorem 4.5, Bouabdallah and Sotteau [2] obtained the following result on

Theorem 4.7 For any and ,

(a) ;

(b) every -graph such that is -regular and diameter and has a minimal routing for which the load of all edges is the same;

(c) if and are odd;

(d) for any and with .

Problem 4.3 was solved for by Bouabdallah and Sotteau [2], who also obtained for any , and for any . Recently, Xu et al [36] have determined if

Theorem 4.8 For any ,

Note the value of has not been determined for

Conjecture 4.9 For any , if

Theorem 4.10 (Xu et al [38]) For any , we have

An asymptotic result on has been given by Heydemann et al [17].

Theorem 4.11 For any given ,

where the upper bound holds for .

4.4. General Results Subject to Degree and Diameter

Theorem 4.12 (Xu et al. [40]) For any connected graph of order and maximum degree ,

Considering a special case of in Theorem 4.12, we obtain the upper bound in (1) immediately.

Theorem 4.13 (Heydemann et al [17]) Let be a graph of order , maximum degree and diameter ,

(a) ,

(b)

Theorem 4.14 (Heydemann et al [17]) If is a graph of order and diameter with no end vertex, then .

Manoussakis and Tuza [26] obtained some upper bounds on the forwarding indies for digraphs subject to degree constraints.

Theorem 4.15 Let be a strongly connected digraph of order and minimum degree . Then

(a) ;

(b) if is sufficient large compared to .

Considering the minimum degree rather the maximum degree , we can propose an analogy of and as follows. Given and , let

However, the problem determining and is simple.

Theorem 4.16 (Xu et al. [40]) For any and with ,

## 5 Difficulty, Methods and Relations to Other Topics

5.1. Difficulty of Determining the Forwarding Indices

As we have stated in Subsection 2.1, the problem of computing the forwarding indices of a general graph is an NP-complete problem. Also, for a given graph , determining its forwarding indices and is also very difficult.

The first difficulty is designing a routing such that for any or for any can be conveniently computed. An ideal routing can be found by the current algorithms for finding shortest paths. However, in general, it is not always the case that the forwarding indices of a graph can be obtained by a minimum routing.

For example, consider the wheel of order seven. The hub , other vertices . A minimum and bidirectional routing is defined as follows.

Then,

Thus, we have .

However, if we define a routing that is the same as the minimum routing except for . Then the routing is not minimum. We have

Thus, we have .

The second difficulty is that the forwarding indices are always attained by a bidirectional routing. For example, for the hypercube , . Since is odd, can not be attained by a bidirectional routing.

5.2. Methods of Determining the Forwarding Indices

To the knowledge of the author, one of the actual methods of determining forwarding index is to compute the sum of all pairs of vertices according to (4) for some Cayley graphs. In fact, the forwarding indices of many Cayley graphs are determined by using (4), for example, the folded cube [21], the augmented cube [37] and so on, list in the next section.

Although Cayley graphs, one class of vertex-transitive graphs, are of hight symmetry, it is not always easy to compute the distance from a fixed vertex to all other vertices for some Cayley graphs. For example, The -dimensional cube-connected cycle , constructed from by replacing each of its vertices with a cycle of length , is a Cayley graph proved by Carlsson et al. [4]. Until now, one has not yet determined exactly its sum of all pairs of vertices, and so only can give its forwarding indices asymptotically (see, Shahrokhi and Székely [29], and Yan et al. [41]).

Unfortunately, for the edge-forwarding index, there is no an analogy of (4). But the lower bound of given in (2) is useful. One may design a routing such that attains this lower bound. For example, the edge-forwarding indices of the folded cube [21] and the augmented cube [37] are determined by this method.

Making use of results on the Cartesian product is one of methods determining forwarding indices. Using Theorem 2.15, Xu et al [35] determined the vertex-forwarding indices and the edge-forwarding indices for the generalized hypercube , the undirected toroidal mesh , the directed toroidal mesh , , all of which can be regarded as the Cartesian products.

5.3. Relations to Other Topics

To the knowledge of the author, until now one has not yet find an approximation algorithm with good performance ratio for finding routings of general graphs. However, de le Vega and Manoussakis [10] showed that the problem of determining the value of the forwarding index (respectively, the forwarding index of minimal routings) is an instance of the multicommodity flow problem (respectively, flow with multipliers). Since many very good heuristics or approximation algorithms are known for these flow problems [1, 23, 24], it follows from these results that all of these algorithms can be used for calculating the forwarding index.

Teranishi [31], Laplacian spectra and invariants of graphs.

## 6 Forwarding Indices of Some Graphs

The forwarding indices of some very particular graphs have been determined. We list all main results that we are interested in and have known, all of which not noted can be found in Heydemann et at [17] or determined easily.

1. For a complete graph , and .

2. For a star , and .

3. For a path , . and

4. For the complete bipartite , and if , and

In particular,

5. For a directed cycle , . For an undirected cycle , and .

6. The -dimensional undirected toroidal mesh is defined as cartesian product of undirected cycles , of order , for . The , denoted by , is called a -ary -cube a generalized -cube. Xu et al [35] determined that

In particular,

The last result was obtained by Heydemann et al [17].

7. The -dimensional directed toroidal mesh is defined as the cartesian product of directed cycles of order , for each . Set . Xu et al [35] determined that

In particular,

8. The -dimensional generalized hypercube, denoted by , where is an integer for each , is defined as the cartesian products . If , then is called the -ary -dimensional cube, denoted by . It is clear that is . Xu et al [35] determined that

In particular,

For the -dimensional hypercube ,

The last result was obtained by Heydemann et al [17].

9. For the crossed cube ,