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Subgraphs and well-quasi-ordering. (English) Zbl 0762.05093
Let $$K$$ be a class, a quasi-ordering $$\preccurlyeq$$ on $$K$$ is said to be a well-quasi-ordering (shortly a wqo) if for every infinite sequence $$x_ 1,x_ 2,\dots$$ of elements of $$K$$ there are $$i,j$$, $$i<j$$, such that $$x_ i\preccurlyeq x_ j$$. For the class $$G$$ of all finite simple graphs neither the subgraph relation $$\subseteq$$ nor the induced subgraph relation $$\preccurlyeq$$ is a wqo.
In the paper ideals $$H$$ of $$G$$ are studied such that $$\subseteq$$ or $$\preccurlyeq$$ is a wqo on $$H$$. The following theorem is proved: Let $$H$$ be an ideal of $$G$$, with respect to $$\subseteq$$, then the following are equivalent: (1) $$(H,\subseteq)$$ is a wqo; (2) $$(H,\preccurlyeq)$$ is a wqo; (3) $$H$$ contains only finitely many $$C_ n$$ and $$F_ n$$, where $$C_ n$$ is a circuit on $$n$$ vertices, $$F_ n$$ is the graph on $$\{y_ 1,y_ 2,z_ 1,z_ 2,x_ 1,x_ 2,\dots,x_ n\}$$ with the edge set $$\{y_ 1x_ 1,y_ 2x_ 1,z_ 1x_ n,z_ 2x_ n,x_ 1x_ 2,x_ 2x_ 3,\dots,x_{n-1}x_ n\}$$.
For $$\preccurlyeq$$ three wqo ideals are presented, one being the class of all graphs without the path on $$n$$ vertices as a subgraph, two others are classes of bipartite graphs.
Connections between the obtained results and digraphs are considered (although no necessary and sufficient condition for an ideal $$D$$ of digraphs for which $$(D,\subseteq)$$ is a wqo is known yet). Examples are constructed disproving some of the possible generalizations of the stated results.
Reviewer: M.Demlová (Praha)

##### MSC:
 05C99 Graph theory 06A06 Partial orders, general
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##### References:
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