AI 中文总结
本文针对某论文提出的是否能对属于某图$H$对应的$\mathrm{obs}^*(H)$的图$G$进行刻画的问题，给出了完整解答。
AI 中文摘要
图$G$到图$H$的全同态是顶点集上的函数，可保留顶点间的邻接与非邻接关系。若图$G$不存在到$H$的全同态，但其每个真顶点诱导子图都存在，则称$G$为极小$H$-障碍图，这类图的顶点数最多为$|V(H)|+1$，顶点数为$|V(H)|+1$的极小$H$-障碍图集合记为$\mathrm{obs}^*(H)$。在Santiago Guzmán-Pro于2024年发表的论文《全同态到路径与环》（《离散数学》第347卷第3期，第113800页）的问题2中，学者提出是否能对属于某图$H$对应的$\mathrm{obs}^*(H)$的图$G$进行刻画，本文针对该问题给出了完整解答。
英文摘要
A full-homomorphism from a graph $G$ to a graph $H$ is a function on vertex sets that preserves adjacency and non-adjacency of vertices. A graph $G$ is called a minimal $H$-obstruction if it has no full-homomorphism to $H$ but every proper vertex induced subgraph of $G$ does. Such graphs can have at most $|V(H)|+1$ vertices. The set of minimal $H$-obstructions on $|V(H)|+1$ vertices is denoted by $\mathrm{obs*}(H)$. In question 2 of the paper "Santiago Guzm{á}n-Pro, Full-homomorphisms to paths and cycles, Discrete Mathematics, 347(3):113800, 2024" it is asked if there is a characterization of those graphs $G$ that lie in $\mathrm{obs*}(H)$ for some graph $H$. In this paper, we give a complete answer to this question.
Comments12 pages, 2 figures. Submitted for publication
