Isomorphisms of graded semiconnected algebras

Let $A_{\bullet }$ and $B_{\bullet }$ be nonnegatively graded algebras, finitely generated in degrees less than $2$ and with semisimple base rings $A_0$ and $B_0$. We prove that if $A\simeq B$ as ungraded algebras, then $A_{\bullet }\simeq B_{\bullet }$ as graded algebras. This generalises a theorem of Bell and Zhang arXiv:1509.08812 in the connected case $A_0=k=B_0$, and the path-algebra version when $A_0$ and $B_0$ are $k$-elementary due to Gaddis arXiv:1712.01650. We also discuss some related open problems for graded algebras.