Linear layouts measuring neighbourhoods in graphs Original Research Article

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 image is the smallest integer k, such that there is a linear layout image, such that for every image there are at most k edges image with image and image. The neighbourhood-width of a graph is the smallest integer k, such that there is a linear layout image, such that for every image the vertices u with image can be divided into at most k subsets each members having the same neighbourhood with respect to the vertices image with image. 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 image and we give several conditions such that graphs of bounded neighbourhood-width have bounded path-width or bounded cut-width.