Zum basisproblem der nicht in die projektive ebene einbettbaren graphen, I

Le Γ be the set of all finite graphs which are not embeddable into the projective plane. We write G₁≺G₂, if G₂ is homomorphic to G₁, that is, if there is a subgraph G′₂ of G₂ such that G₁ can be obtained from G′₂ by contraction of edges in G′₂. The subset of all minimal graphs in the partially ordered set Γ_≺ is called the minimal basis of Γ. We denote this basis by M(Γ). Every graph of M(Γ) is called a minimal graph (of Γ). In a former paper [2], we determined all disconnected minimal graphs and all minimal graphs separable by one or two vertices. Hence, we need only consider three-connected minimal graphs. Let us say that in our Proposition 7 of [2] we omitted a bracket in G₁₄. In this graph, namely, the uppermost 2 must be replaced by <2>.

In the following we deal with two new cases, in which we succeed in determining the minimal graphs:

First, let us suppose that G∈M(Γ) has the properties: (1) G is (at least) three-connected and (2) there is a vertex a in G such that the graph G/a (that is, one has to destroy the vertex a and all edges of G incident to a) contains a topologic K₅ but there exists no topologic K_3,3 contained in G/a. We prove the theorem that there are exactly two graphs in M(Γ) satisfying (1) and (2). One of these graphs is the G₁₄. The other graph, denoted by D, can be obtained from the K₅ where we let T be a triple of neighboring edges in the K₅, by inserting a new vertex on every edge of T and by then connecting these three new vertices through three edges with one further new vertex a lying outside the topologic K₅ (see Figure 2).

Second, we consider non-planar graphs G with the property that G/a is planar for every vertex a of G. In another paper [3] these graphs have been called nearly planar. Our next question is: Are there nearly planar graphs not being embeddable into the projective plane? We prove the theorem that all nearly planar graphs can be embedded into the projective plane with a single exception of one graph F. This exceptional graph F consists of two disjoint K₄ and four edges (a_i, b_i), i=1,…, 4 which join the vertices a_i of the one K₄ with the vertices b_i of the other K₄. We then show that this graph is a minimal graph of Γ. So, in the problem of determining the minimal basis of Γ, the case remains that one can assume: (1) There is a vertex a in G such that G/a contains a topologic K_3,3 and (2) G is three-connected.