This paper introduces the notion of an $(ι,q)$-critical graph. The isolation number of a graph $G$, denoted by $ι(G)$ and also known as the vertex-edge domination number of $G$, is the size of a smallest subset $D$ of the vertex set of $G$ such that the subgraph induced by the set of vertices that are not in the closed neighbourhood of $D$ has no edges. A graph $G$ is $(ι,q)$-critical if every subdivision of $q$ edges of $G$ gives a graph whose isolation number is greater than $ι(G)$, and $G$ has $q-1$ edges such that subdividing them gives a graph whose isolation number is $ι(G)$. We show that an $(ι,q)$-critical graph exists for every integer $q \ge 1$. We prove that if $G$ is a connected $m$-edge non-star graph, then $G$ is $(ι,q)$-critical for some $q \le m - 1$. We show that this bound is best possible. We provide a general characterization of $(ι,1)$-critical graphs as well as a constructive characterization of $(ι,1)$-critical trees, demonstrating that $(ι,1)$-criticality can be checked in linear time for trees.

翻译：本文引入了$(ι,q)$-临界图的概念。图$G$的孤立数，记为$ι(G)$，也称为图的顶点边支配数，是指顶点集$V(G)$的最小子集$D$的大小，使得不在$D$的闭邻域中的顶点所诱导的子图不含边。如果对$G$的$q$条边进行细分后得到的图其孤立数均大于$ι(G)$，且存在$q-1$条边使得细分这些边后得到的图其孤立数等于$ι(G)$，则称图$G$是$(ι,q)$-临界图。我们证明了对任意整数$q \ge 1$均存在$(ι,q)$-临界图。同时证明，若$G$是连通的具有$m$条边的非星图，则对于某个$q \le m - 1$，$G$是$(ι,q)$-临界图，并证明该界是紧的。我们给出了$(ι,1)$-临界图的一般刻画，以及$(ι,1)$-临界树的构造性刻画，从而证明对于树而言，$(ι,1)$-临界性可在线性时间内判定。