This paper introduces the notion of $(ι,q)$-critical graphs. The isolation number of a graph $G$, denoted by $ι(G)$ and also known as the vertex-edge domination number, is the minimum number of vertices in a set $D$ such that the subgraph induced by the vertices not in the closed neighbourhood of $D$ has no edges. A graph $G$ is $(ι,q)$-critical, $q \ge 1$, if the subdivision of any $q$ edges in $G$ gives a graph with isolation number greater than $ι(G)$ and there exists a set of $q-1$ edges such that subdividing them gives a graph with isolation number equal to $ι(G)$. We prove that for each integer $q \ge 1$ there exists a $(ι,q)$-critical graph, while for a given graph $G$, the admissible values of $q$ satisfy $1 \le q \le |E(G)| - 1$. In addition, we provide a general characterisation of $(ι,1)$-critical graphs as well as a constructive characterisation of $(ι,1)$-critical trees.

翻译：本文引入了$(ι,q)$-临界图的概念。图$G$的隔离数（记为$ι(G)$，亦称顶点-边支配数）是指满足以下条件的最小顶点集$D$的基数：不在$D$的闭邻域中的顶点所诱导的子图不含边。若对图$G$中任意$q$条边进行细分后所得图的隔离数均大于$ι(G)$，且存在$q-1$条边使得细分后所得图的隔离数等于$ι(G)$，则称$G$为$(ι,q)$-临界图（$q \ge 1$）。我们证明了对任意整数$q \ge 1$均存在$(ι,q)$-临界图，而对于给定图$G$，其允许的$q$值满足$1 \le q \le |E(G)| - 1$。此外，本文给出了$(ι,1)$-临界图的一般性刻画，并建立了$(ι,1)$-临界树的构造性刻画。