Graphs containing finite induced paths of unbounded length
Maurice Pouzet, Imed Zaguia
备注：
28 pages, 3 figures
发表信息：
Discrete Mathematics & Theoretical Computer Science, vol. 23 no. 2, special issue in honour of Maurice Pouzet, Special issues (March 8, 2022) dmtcs:6915
本文定义了包含无界长度有限诱导路径的 path-minimal 图，并基于一致递归序列和标记图的字典序和，构造了 $2^{\aleph_0}$ 个 age 两两不可比且良拟序的此类图。
摘要：
图 $G$（无向且无环）的 age $\mathcal{A}(G)$ 是 $G$ 的有限诱导子图的集合，按同构考虑并由可嵌入性排序。如果不包含无限反链，则它对该序是良拟序（wqo）。如果图包含无界长度的有限诱导路径，且具有此性质的每个诱导子图 $G'$ 都嵌入 $G$，则该图是 path-minimal 的。我们构造了 $2^{\aleph_0}$ 个 path-minimal 图，它们的 age 在集合包含下两两不可比，且都是 wqo。我们的构造基于一致递归序列和标记图的字典序和。
英文摘要：
The age $\mathcal{A}(G)$ of a graph $G$ (undirected and without loops) is the collection of finite induced subgraphs of $G$, considered up to isomorphy and ordered by embeddability. It is well-quasi-ordered (wqo) for this order if it contains no infinite antichain. A graph is \emph{path-minimal} if it contains finite induced paths of unbounded length and every induced subgraph $G'$ with this property embeds $G$. We construct $2^{\aleph_0}$ path-minimal graphs whose ages are pairwise incomparable with set inclusion and which are wqo. Our construction is based on uniformly recurrent sequences and lexicographical sums of labelled graphs.
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HUN-REN Alfréd Rényi Institute of Mathematics（HUN-REN阿尔弗雷德·雷尼数学研究所）；ELTE Eötvös Loránd University（布达佩斯厄特沃什·罗兰大学）；Department of Mathematics, University of California at San Diego（加州大学圣地亚哥分校数学系）
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