In this paper, we investigate algorithms for some related graph parameters. Each of these asks for a linear ordering of the vertices of the graph (or can be formulated as such), and there are constructive linear time algorithms for the fixed parameter versions of the problems. Examples are cutwidth, pathwidth, and directed or weighted variants of these. However, these algorithms have complicated technical details. This paper attempts to present ideas in these algorithms in a different more easily accessible manner, by showing that the algorithms can be obtained by a stepwise modification of a trivial hypothetical non-deterministic algorithm. The methodology is applied for a generalisation of the cutwidth problem to weighted mixed graphs. As a consequence, we obtain new algorithmic results for problems like modified cutwidth, and rederive known results for other related problems with simpler proofs.