Topological Bandwidth Calculator

bandwidth via subdivision

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About Topological Bandwidth Calculator

A topological bandwidth calculator computing bw_t(G): subdivide edges (add new vertices), then minimize the bandwidth. Always bw_t ≤ bw. Captures 'smoothed' layout complexity. For trees: bw_t = ⌈(Δ+1)/2⌉ where Δ is max degree. Client-side.

Topological Bandwidth Calculator Features

  • bw_t value
  • vs bw
  • Subdivision
  • Tree formula
  • Common graphs
Topological bandwidth bw_t(G): minimum bandwidth achievable by subdividing edges. Add degree-2 vertices to spread out edges. bw_t ≤ bw always. For trees: bw_t(T) = ⌈(Δ(T)+1)/2⌉. Captures embedding complexity independent of edge lengths.

How to Use

Select graph:

  • bw_t: Topological bw
  • bw: Compare with bw
  • Δ: Max degree

Tree Formula

For trees: bw_t(T) = ⌈(Δ(T)+1)/2⌉. Beautiful closed form! Depends only on max degree. For general graphs: NP-hard. Subdivision 'amortizes' high-bandwidth edges by spreading them out.

Theory

bw_t relates bandwidth to topological properties. Adding subdivisions smooths out edge congestion. Connection to cutwidth: cw(G) ≥ bw_t(G). For planar graphs: bw_t = O(√n). Deep structural relationships.

Step-by-Step Instructions

  1. 1Select graph.
  2. 2Compute bw.
  3. 3Subdivide edges.
  4. 4Minimize bw.
  5. 5Compare bw_t vs bw.

Topological Bandwidth Calculator — Frequently Asked Questions

Why subdivide edges?+

Subdivision adds degree-2 vertices on edges, effectively 'stretching' them. High-bandwidth edges get broken into shorter segments. This can dramatically reduce maximum stretch. Like adding relay stations.

Is bw_t always less than bw?+

bw_t ≤ bw always (subdivisions can only help or maintain bandwidth). For K_4: bw=2, bw_t=2 (can't improve). For stars K_{1,n}: bw=⌊n/2⌋, bw_t=⌈(n+1)/2⌉ (similar). Big gap possible for dense graphs.

What's the connection to cutwidth?+

cutwidth(G) ≥ bw_t(G). Cutwidth upper-bounds topological bandwidth. Both measure layout complexity but cutwidth counts edge congestion while bw_t measures edge stretch after smoothing.

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