Conjecture on regularity and component structure of distinguishing critical graphs

Let $d$ be a positive integer and let $G$ be a graph. The graph $G$ is $d$-distinguishing critical if its distinguishing number is $d$ and every proper induced subgraph of $G$ has distinguishing number less than $d$.

Regularity and component conjecture. (i) If $G$ is a $d$-distinguishing critical graph, then $G$ is a $k$-regular graph for some $k\leq d$. (ii) If $G$ is a disconnected $d$-distinguishing critical graph, then each component of $G$ is a complete graph.

The paper establishes this structure for several small values of $d$ and proves that the only distinguishing critical tree is $K_2$; the conjecture proposes the general regularity and complete-component conclusions.