# Topological bandwidth

## 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## Preview

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