OBSTRUCTION SETS FOR OUTER--PROJECTIVE--PLANAR GRAPHS Dan Archdeacon, Nora Hartsfield, C.H.C. Little, and Bojan Mohar Summary: OBSTRUCTION SETS FOR OUTER--PROJECTIVE--PLANAR GRAPHS Dan Archdeacon, Nora Hartsfield, C.H.C. Little, and Bojan Mohar University of Vermont, Western Washington University, Massey University, and the University of Ljubljana Abstract. A graph G is outer­projective­planar if it can be embedded in the projective plane so that every vertex appears on the boundary of a single face. We exhibit obstruction sets for outer­projective­ planar graphs with respect to the subdivision, minor, and Y \Delta orderings. Equivalently, we find the minimal non­outer­projective­planar graphs under these orderings. x1 Introduction The most frequently cited [B] result in graph theory is Kuratowski's Theorem [K], which states that a graph is planar if and only if it does not contain a subdivision of either K 5 or K 3;3 . This is an example of an obstruction theorem; a characterization of graphs with a particular property in terms of excluded subgraphs. Obstruction theorems may involve other properties besides planarity and other orderings besides the subgraph order. Let P be a property of graphs, formally, P is some collection of graphs. Let ¯ be a partial ordering on all graphs. We say that P is hereditary under ¯ if G 2 P and H ¯ G implies that H 2 P . For example, the collection of all planar graphs is hereditary under the subgraph ordering. An obstruction for P under ¯ is a graph G such that G = 2 P , but H 2 P for all H OE G (here ``OE'' means ``¯'' but not equal). In words, an obstruction is a minimal graph Collections: Mathematics