# Embedding onto Wheel-like Networks

## TL;DR

This paper analyzes the graph embedding problem for wheel-like networks, providing exact bounds and calculations for dilation, congestion, and wirelength, which are crucial for efficient interconnection network simulation.

## Contribution

It computes exact dilation, congestion, and wirelength for embeddings onto wheel-like networks, establishing sharp bounds and extending results to various complex graph classes.

## Key findings

- Exact dilation of embedding wheel-like networks into hypertrees
- Exact congestion of embedding windmill graphs into circulant graphs
- Estimated wirelength of embedding wheels and fans into multiple graph classes

## Abstract

One of the important features of an interconnection network is its ability to efficiently simulate programs or parallel algorithms written for other architectures. Such a simulation problem can be mathematically formulated as a graph embedding problem. In this paper we compute the lower bound for dilation and congestion of embedding onto wheel-like networks. Further, we compute the exact dilation of embedding wheel-like networks into hypertrees, proving that the lower bound obtained is sharp. Again, we compute the exact congestion of embedding windmill graphs into circulant graphs, proving that the lower bound obtained is sharp. Further, we compute the exact wirelength of embedding wheels and fans into 1,2-fault hamiltonian graphs. Using this we estimate the exact wirelength of embedding wheels and fans into circulant graphs, generalized Petersen graphs, augmented cubes, crossed cubes, M\"{o}bius cubes, twisted cubes, twisted $n$-cubes, locally twisted cubes, generalized twisted cubes, odd-dimensional cube connected cycle, hierarchical cubic networks, alternating group graphs, arrangement graphs, 3-regular planer hamiltonian graphs, star graphs, generalised matching networks, fully connected cubic networks, tori and 1-fault traceable graphs.