Algorithms and obstructions for linear-width and related search parameters

The linear-width of a graph G is de ned to be the smallest integer k such that the edges of G can be arranged in a linear ordering e1;:::;er in such a way that for every i = 1;:::;r , 1, there are at most k vertices incident to edges that belong both to fe1;:::;eig and to fei+1;:::;er g. In this paper, we give a set of 57 graphs and prove that it is the set of the minimal forbidden minors for the class of graphs with linear-width at most two. Our proof alsogives a linear timealgorithmthat either reports that a given graph has linear-width more than two or outputs an edge ordering of minimum linear-width. We further prove a structural connection between linear-width and the mixed search number which enables us to determine, for any k 1, the set acyclic forbidden minors for the class of graphs with linear-width k. Moreover, due to this connection, our algorithm can be transfered to two linear time algorithms that check whether a graph has mixed search or edge search number at most two...

• Type: Journal
• Year: 2000
• Where: DAM
• Authors: Dimitrios M. Thilikos
• Keywords: DAM 2000, Forbidden Minors, Minimal Forbidden Minors, Smallest Integer