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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

  

Source: Archdeacon, Dan - Department of Mathematics and Statistics, University of Vermont

 

Collections: Mathematics