The complexity of restricted star colouring

Restricted star colouring is a variant of introduced to design heuristic algorithms estimate sparse Hessian matrices. For $k\in\mathbb{N}$, $k$-restricted ($k$-rs colouring) graph $G$ function $f:V(G)\to{0,1,\dots,k-1}$ such that (i)$f(x)\neq f(y)$ for every edge $xy$ G, and (ii) there no bicoloured 3-vertex path ($P_3$) in with the higher colour on its middle vertex. We show $k\geq 3$, it NP-complete test whether given planar bipartite maximum degree $k$ arbitrarily large girth admits $k$-rs colouring, thereby answer problem posed by Shalu Sandhya (Graphs Combinatorics, 2016). In addition, 3-star colourable 3-rs colouring. also prove all $\epsilon > 0$, optimization restricted 2-degenerate minimum number colours NP-hard approximate within $n^{(1/3)-\epsilon}$. On positive side, we (i) linear-time algorithm colourability trees, an $O(n^3)$-time chordal graphs.