The forwarding indices of graphs - a survey

Jun-Ming Xu, Min Xu
2013 Opuscula Mathematica  
A routing R of a connected graph G of order n is a collection of n(n − 1) simple paths connecting every ordered pair of vertices of G. The vertex-forwarding index ξ(G, R) of G with respect to a routing R is defined as the maximum number of paths in R passing through any vertex of G. The vertex-forwarding index ξ(G) of G is defined as the minimum ξ(G, R) over all routings R of G. Similarly, the edge-forwarding index π(G, R) of G with respect to a routing R is the maximum number of paths in R
more » ... ing through any edge of G. The edge-forwarding index π(G) of G is the minimum π(G, R) over all routings R of G. 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 for over twenty years. This paper surveys some known results on these forwarding indices, further research problems and several conjectures, also states some difficulty and relations to other topics in graph theory. 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 the situation. The network designer must specify a set of routes for each pair (x, y) of vertices in advance, indicating a fixed route which carries the data transmitted from the message source x to the destination y. Such a choice of routes is called a routing. We follow [61] for graph-theoretical terminology and notation not defined here. A graph G = (V, E) always means a simple and connected graph, where V = V (G) is the vertex-set and E = E(G) is the edge-set of G, |V | is the order of G and |E| is the size of G. It is well known that the underlying topology of a communication network can be modeled by a connected graph G = (V, E), where V is the set of processors and E is the set of communication links in the network. Let G be a connected graph of order n. A routing R in G is a set of n(n − 1) fixed paths for all ordered pairs (x, y) of vertices of G. The path R(x, y) specified by R carries the data transmitted from the source x to the destination y. A routing R in G is said to be minimal, denoted by R m , if each of the paths specified by R is shortest; symmetric or bidirectional if for all vertices x and y, the path R(y, x) is the reverse of the path R(x, y) specified by R; consistent if for any two vertices x and y, and for each vertex z belonging to the path R(x, y) specified by R, the path R(x, y) is the concatenation of the paths R(x, z) and R(z, y). It is possible that the fixed paths specified by a given routing R going through some vertex are too many, which means that the routing R loads the vertex too much. The load of any vertex is limited by the capacity of the vertex, for otherwise it would affect the efficiency of transmission and even result in the 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, in 1987, Chung et al. [16] proposed the concept of the forwarding index. Let R(G) and R m (G) be the sets of routings and minimum routings in a graph G, respectively. For a given R ∈ R(G) and x ∈ V (G), the load of x with respect to R, denoted by ξ x (G, R), is defined as the number of paths specified by R going through x. The parameter is called the forwarding index of G with respect to R, and the parameter Similar problems were studied for edges by Heydemann et al. [32] in 1989. The load of an edge e with respect to R, denoted by π e (G, R), is defined as the number of the paths specified by R which go through it. The edge-forwarding index of G with respect to R, denoted by π(G, R), is the maximum number of paths specified by R going through any edge of G, i.e., π(G, R) = max{π e (G, R) : e ∈ E(G)}, The forwarding indices of graphs -a survey 347 and the edge-forwarding index of G is defined as π(G) = min{π(G, R) : R ∈ R m (G)}. For the minimal routing R m , let ξ m (G) = min{ξ(G, R m ) : R m ∈ R m (G)} and π m (G) = min{π(G, R m ) : R m ∈ R m (G)}. Clearly, ξ(G) ≤ ξ m (G) and π(G) ≤ π m (G).
doi:10.7494/opmath.2013.33.2.345 fatcat:47zeik3anjam5ghhxyqdiehm4e