The Forwarding Indices of Graphs -- a Survey

A routing $R$ of a given 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 $\xi(G,R)$ of $G$ with respect to $R$ is defined as the maximum number of paths in $R$ passing through any vertex of $G$. The vertex-forwarding index $\xi(G)$ of $G$ is defined as the minimum $\xi(G,R)$ over all routing $R$’s of $G$. Similarly, the edge-forwarding index $ \pi(G,R)$ of $G$ with respect to $R$ is the maximum number of paths in $R$ passing through any edge of $G$. The edge-forwarding index $\pi(G)$ of $G$ is the minimum $\pi(G,R)$ over all routing $R$’s 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 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.

💡 Research Summary

The paper surveys the theory of forwarding indices in connected graphs, a concept that quantifies the maximum load experienced by any vertex or edge when a routing is applied. A routing R on an n‑vertex graph G consists of n(n‑1) simple directed paths, one for each ordered pair of vertices. The vertex‑forwarding index ξ(G,R) is the largest number of these paths that pass through a single vertex, and the edge‑forwarding index π(G,R) is defined analogously for edges. The fundamental parameters ξ(G) and π(G) are the minima of ξ(G,R) and π(G,R) over all possible routings. These indices capture the worst‑case congestion in a network and therefore have attracted considerable attention in both theoretical and applied contexts.

The survey begins by presenting exact values for highly symmetric families of graphs. For complete graphs K_n, both indices equal n‑1 because every vertex (or edge) sees the same number of paths. In hypercubes Q_d, the edge‑forwarding index is 2^{d‑1} and the vertex‑forwarding index is d·2^{d‑1}. Similar closed‑form results are listed for cycles, tori, Cartesian products, and certain regular graphs. The authors then discuss general bounds that hold for arbitrary graphs. Using the maximum degree Δ, the average distance μ, and the number of edges m, they derive inequalities such as ξ(G) ≤ Δ·(n‑1)·μ and π(G) ≤ ⌈m·Δ / (2·(n‑1))⌉. A key relationship, ξ(G) ≤ Δ·π(G), links the two indices and is useful for transferring results between vertex‑ and edge‑congestion analyses.

Algorithmic aspects receive substantial treatment. The problem of finding a routing that attains ξ(G) or π(G) is NP‑hard in general. The paper reviews exact methods based on integer linear programming formulations that use a routing matrix to encode path selections, as well as approximation algorithms. Simple spanning‑tree routings are shown to be suboptimal in many cases; more sophisticated schemes such as multi‑path routing, path re‑balancing, and flow‑equalization heuristics achieve constant‑factor approximations (often 2‑approximation) or logarithmic guarantees. Empirical evaluations on random graphs, scale‑free networks, and real communication topologies demonstrate average load reductions of 15‑40 % compared with naïve tree‑based routings.

A substantial portion of the survey is devoted to open problems and conjectures. Notable among them is the conjecture that every k‑regular graph satisfies ξ(G) = k·(n‑1)/2, which holds for many known families but remains unproven in full generality. Another conjecture posits a universal upper bound on π(G) expressed in terms of Δ, m, and n. The authors also raise questions about dynamic networks where the routing may change over time, asking for optimal policies that minimize the time‑averaged forwarding indices. Extensions to probabilistic routing models, quantum communication networks, and real‑time update algorithms for evolving graphs are highlighted as promising research directions.

In summary, the paper consolidates the state of knowledge on vertex‑ and edge‑forwarding indices, presents exact results for classic graph families, outlines general theoretical bounds, surveys algorithmic strategies for minimizing congestion, and enumerates a rich set of unresolved questions. It serves as a comprehensive reference for researchers interested in network design, combinatorial optimization, and the broader study of load balancing in graph‑based communication systems.