Minimizing width in linear layouts

Abstract

A (linear) layout of an undirected graph G is a one-to-one function mapping the vertices of G to integers. The cutwidth of G under a linear layout L, denoted by cw(G,L), is the maximum, taken over all possible i, of the number of edges connecting vertices assigned to integers less than i to vertices assigned to integers at least as large as i. The cutwidth of a graph G, denoted by cw(G), is the minimum of cw(G,L), taken over all possible linear layouts L. The problem of determining the cutwidth of a graph, called the Min Cut Linear Arrangement problem, has applications in VLSI, for example in the minimization of interconnection channels in Weinberger arrays [16].

We describe a relationship between the cutwidth of a graph G and its “search number” [10], denoted by s(G). We show that, for all graphs G, s(G) ≤ cw(G) ≤ ≫deg(G)/ 2⌋ · s(G), where deg(G) denotes the maximum degree of any vertex in G. In particular, this means that, for any graph G with maximum vertex degree 3, s(G)=cw(G).

The earlier dynamic programming algorithm of Gurari and Sudborough [5 ] is improved to show that, for any k≥2, the problem of deciding if a given graph has cutwidth at most k can be done in O(n^k−1) steps. We also characterize the classes of graphs with cutwidth 2 and cutwidth 3. The latter characterization strongly suggests a linear time algorithm to determine whether a given graph has cutwidth at most three.

A portion of this work was completed while at the National Technical University of Athens, in Athens, Greece

supported in part by NSF Grant number MCS 81-09280