For any set $X$, we let $[X]^2 = \big\{\{x,y\}:x\neq y \in X\big\}$.


If $G=(V,E)$ is a simple, undirected graph, and $v\in V$, let $N(v) = \{z\in V: \{v,z\}\in E\}$. Given any $v\in V$, we use the following notation:


1. Let $G\setminus\{v\} := (V\setminus \{v\}, E \cap [V\setminus\{v\}]^2)$, and
2. if $w\in V$ with $\{v,w\}\in E$, let $(G\setminus\{v\})^w := (V\setminus\{v\}, E\_{v,w})$ where $E\_{v,w}:= (E \cap [V\setminus\{v\}]^2) \cup \{\{w,z\}: z\in N(v)\}$.


A graph $G=(V,E)$ is said to be *vertex-critical* if $\chi(G\setminus\{v\}) < \chi(G)$ for every $v\in V$. We say $G$ is *collapse-critical* if $\chi\big((G\setminus\{v\})^w\big) < \chi(G)$ for all $v\in V$ and $w\in V$ such that $\{v,w\}\in E$. It is clear that $\chi(G\setminus\{v\})\leq \chi\big((G\setminus\{v\})^w\big)$, so collapse-criticality implies vertex-criticality.


**Question.** What is an example of a finite, connected graph that is vertex-critical, but not collapse-critical?



Two kinds of vertex-criticality
Unless I'm mistaken, your definition of $(G\setminus \{v\})^w$ is symmetrical, and generally referred as an edge-contraction, so collapse critical is the same as edge-contraction critical.


[this graph](https://math.stackexchange.com/questions/2430688/vertex-critical-graph-with-at-least-one-non-critical-edge) is vertex-critical, but not collapse-critical.


[duplicate](https://math.stackexchange.com/questions/2580899/distinguishing-vertex-edge-and-edge-contraction-critical-graphs)


