CAAP 1983: CAAP'83 pp 317-331

# Topological bandwidth

• F. S. Makedon
• I. H. Sudborough
Contributed Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 159)

## Abstract

Let L be a one-to-one function mapping the vertices of an undirected graph G to the positive integers. (L is called a "linear layout" of G.) The "bandwidth of G under the layout L", denoted by b (G,L), is the maximum difference between the integers assigned to vertices in G connected by an edge. The "bandwidth of G", denoted b(G), is the min { b(G,L) | L is a linear layout of G }. The "topological bandwidth of G", denoted by tb(G), is the minimum bandwidth of all graphs obtained from G by subdividing some of G's edges with some number (possibly zero) of degree two vertices.

The "modified cutwidth of G under a layout L", denoted by mcw(G,L), is the maximum, over all integers i, of the number of edges connecting vertices that are assigned to integers smaller than i with vertices assigned to integers larger than i. The "modified cutwidth of G", denoted by mcw(G), is min{ mcw(G,L) | L is a linear layout of G }.

It is shown that, for all graphs G, tb(G)≤mcw(G)+1 and that, for all degree 3 graphs G, tb(G)=mcw(G)+1. This yields an 0(n log n) algorithm for determining the topological bandwidth of an arbitrary degree three tree, using the algorithm in [23] for modified cutwidth. Topological bandwidth is also equated with the "node search number" for degree three graphs.

We give a recursive characterization theorem for topological bandwidth in degree three trees. This yields a description of the set of smallest degree three trees having topological bandwidth k, for all k≥1, and a characterization, by forbidden subtrees, of topological bandwidth k in degree three trees, for all k≥1.

We show that the Topological Bandwidth problem, the Modified Cutwidth problem, the Min Cut Linear Arrangement problem, the Search Number problem, and the Node Search Number problem are all NP-complete even when restricted to graphs that have maximum vertex degree three.

Finally, we give a characterization of graphs with topological bandwidth 2 and thereby derive a linear time algorithm to determine if a graph G satisfies tb(G)=2.

## Keywords

Positive Instance Linear Time Algorithm Connection Point Search Number Biconnected Component
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• F. S. Makedon
• 1