Free Minor Closed Classes and the Kuratowski theorem

Free-minor closed classes [2] and free-planar graphs [3] are considered. Versions of Kuratowski-like theorem for free-planar graphs and Kuratowski theorem for planar graphs are considered. We are using usual definitions of the graph theory [1]. Considering graph topologically and Kuratowski theorem, we use the notion of minor following the theory of Robertson and Seymour[2]. We say, that a graph G is a minor of a graph H, denoting it by G ≺ H, if G can be obtained from H by edge contractions from a subgraph of H, i.e. G can be obtained by vertex deletions, edge deletions and edge contractions from H. A class of graphs A is minor closed, if from G ∈ A and H ≺ G follow that H ∈ A. The set of forbidden minors of a class A is denoted by F (A) which is equal to b{G | G 6∈ A}c, where bBc contains only minimal minors of B: bBc , {G | H ∈ B ∧H ≺ G ⇒ H ∼= G}. N◦(B) denotes the minor closed class with B as its set of forbidden minors, i.e. N◦(B) , {G | ∀H ∈ B : H 6≺ G}. In other words, we may say, that N◦(B) is a minor closed class generated by its forbidden minors in B. For example, N◦(K5,K3,3) is the class of planar graphs, as it is stated by Kuratowski theorem. Another interesting example is free-planar graphs [3]. A planar graph is called free-planar, if after adding an arbitrary edge it remains to be planar. In [3] without a proof is acclaimed, that the class of free-planar graphs is equal to N◦(K− 5 ,K − 3,3), and its characterization in terms of the permitted 3-connected components is given. In this paper we give a proof of this characterization. In [2] a generalization of the notion of free-planar graphs is suggested. We denote by Free(A) the class of graphs that consist of all graphs which should belong to A after adding an arbitrary edge to them. It is easy to see, that, if A is minor closed, then Free(A) is minor closed too [2]. Because of this we use to say, that Free(A) is free-minor-closed-class for a minor closed class A. In [2] Kratochv́ıl proved a theorem: F (Free(A)) = bF (A)− ∪ F (A) ̄c, where B− , {G − e | G ∈ B, e ∈ E(G)} and B ̄ , {H | H ∼= G ̄ v,G ∈ B, v ∈ V (G)} and operation ̄ [in its application G ̄ v] denotes a non unique splitting of vertex v in G, which is the opposite operation to edge addition and its contraction [in result giving vertex v]. We may formulate the unproved statement of [3] as a theorem for class of planar graphs Planar: Theorem 1. Free(Planar) = N◦(K− 5 ,K − 3,3). It is convenient to call the graphs K− 5 ,K − 3,3 – reduced Kuratowski subgraphs (or minors or graphs). Now, direct application of the theorem of Kratochv́ıl gives the proof of theorem 1, that has been already shown in [2]. All possible graphs obtainable following the theorem are in fig. 1. In [2] Kratochv́ıl suggested to prove Kuratowski’s theorem from its weaker version for freeplanar graphs. We do this here in two ways. One way – first specifying the class generated ∗This research is supported by the grant 97.0247 of Latvian Council of Science. †Author’s address: Institute of Mathematics and Computer Science, University of Latvia, 29 Rainis blvd., Riga, Latvia. [email protected]