Linear layouts measuring neighbourhoods in graphs

In this paper we introduce the graph layout parameter neighbourhood-width as a variation of the well-known cut-width. The cut-width of a graph G = (V, E) is the smallest integer k, such that there is a linear layout phi : V -> {1,..., vertical bar V vertical bar}, such that for every 1 <= i < vertical bar V vertical bar there are at most k edges {u, v} with phi(u) <= i and phi(v) > i. The neighbourhood-width of a graph is the smallest integer k, such that there is a linear layout phi, such that for every 1 <= i < vertical bar V vertical bar the vertices u with phi(u) <= i can be divided into at most k subsets each members having the same neighbourhood with respect to the vertices v with phi(v) > i. We show that the neighbourhood-width of a graph differs from its linear clique-width or linear NLC-width at most by one. This relation is used to show that the minimization problem for neighbourhood-width is NP-complete. Furthermore, we prove that simple modifications of neighbourhood-width imply equivalent layout characterizations for linear clique-width and linear NLC-width. We also show that every graph of path-width k or cut-width k has neighbourhood-width at most k+2 and we give several conditions such that graphs of bounded neighbourhood-width have bounded path-width or bounded cut-width. (c) 2006 Elsevier B.V. All rights reserved.