Fig. 1. Selection of zone side length.

Fig. 2. The Fixed zoning process in both (a) Homogeneous networks and (b) Heterogeneous networks.

Fig. 3. (a) Inter-zone communication and (b) Intra-zone communication in VGA.

Fig. 4. Example of the split and merge processes in heterogeneous MANETs.

Fig. 5. Cluster head election algorithm in a certain zone z.

Fig. 6. The zoning process with corresponding virtual topologies for VGA, D-VGA, and optimal cases.

Fig. 7. (a) Diagonal routing capability option and (b) Probability that D-VGA succeeds given that VGA fails.

Fig. 8. On-demand packet forwarding/routing over VGA: an example.

Fig. 9. The illustration of local path restoration in VGA.

Fig. 10. CHs cardinality; homogeneous MANETs.

Fig. 11. CHs cardinality; heterogeneous MANETs.

Fig. 12. Average path length; VGA and D-VGA.

Fig. 13. Communication Overhead in VGA and D-VGA.

Fig. 14. Packet delivery ratio vs. offered load.

Fig. 15. Network end-to-end delay vs. offered load.

Fig. 16. Percentage of broken routes due to mobility.

Fig. 17. Call acceptance rate in the two routing techniques.

Fig. 18. Network control overhead versus offered load using the TC and OD techniques.

Fig. 19. Packet delivery ratio versus offered load under different CH election strategies.

Fig. 20. Effect of power control scheme of VGA on node lifetime.

Fig. 21. OD-based routing scalability as node density increases.

Fig. 22. Figure used in the proof of Proposition 1.

System model notations

Summary of Intra-zone and Inter-zone communication ranges and RTP values

Comparison of clusterhead cardinality for VGA, D-VGA, and optimal clustering

The ILP variables in heterogeneous MANETs

The ILP formulation for finding MCDS

Performance comparison between different algorithms

Here opt is the size of the optimal MCDS; Δ is the maximum node degree; C is the size of the generated connected dominating set; Z is the number of zones in VGA; V, E: the number of nodes and number of edges in the virtual network graph, respectively.