Partial tracks, characterizations and recognition of graphs with path-width at most two

Nancy G. Kinnersley and Michael A. Langston has determined [3] the excluded minors for the class of graphs with path-width at most two. Here we give a simpler presentation of their result. This also leads us to a new characterization, and a linear time recognition algorithm for graphs width path-width at most two. 1 History and introduction Based on the seminal work of Seymour and Robertson [4], it is easy to see, that any class of graphs, which is closed for taking minors, can be characterized by a finite list of excluded minors. There are several examples for such a forbidden minor characterization. The most famous example is Kuratowski’s theorem, which says that a graph is planar iff it has no minor isomorphic to K5 or K3,3. There are many known minor closed classes of graphs. One can easily obtain such a class by bounding a minor-monotone graph parameter, e.g. the tree-width, the path-width, or the Colin de Verdière number of the graphs. Unfortunately there are not too many forbidden minor characterization theorems are known. The reason is the following. Even in the case of considerable simple classes, the number of forbidden minors can be quite large. E.g. Sanders in his Ph.D. thesis determined more than 75 minimal forbidden minors for tree-width at most four, but the list was still incomplete. There are further lower bounds on the length of the list of the forbidden minors for other classes (e.g. [6]). The class of graphs with path-width at most two is a natural class to study. It is quite simple, hence there is a hope to handle its characterization and it is linked to the matrix layout problem (with three tracks). This author’s research was supported by OTKA Grants F.026049 and F.030737 The second author’s research was supported by OTKA Grant T.030074