Degree-truncated DP-colouring of ₄,K2,t\-minor-free graphs

A graph G is degree-truncated DP-k-colourable (respectively, degree-truncated k-choosable) if it is DP-f-colourable (respectively, f-choosable), where f(v)=min,dG(v)\ for every vertex v. Hutchinson proved that every $2$-connected maximal outerplanar graph other than the triangle is degree-truncated $5$-choosable. Lo, Wang, Zhou and Zhu strengthened this result by proving that every $2$-connected K2,4-minor-free graph other than cycles and complete graphs is degree-truncated DP-$5$-colourable. Outerplanar graphs are precisely the ₄,K2,3\-minor-free graphs. We study how the truncation threshold grows when K2,3 is replaced by K2,t, while K₄-minor remains excluded. A bound independent of t is impossible, as shown by the family 2,n| n≥ 2\. Let Sₜ be the class of $2$-connected ₄,K2,t\-minor-free graphs other than cycles. Writing χ^dDP( Sₜ) for the least truncation threshold valid throughout this class, we prove that, for every integer t≥3, \[ 2√2t ≤ χ^{{{d}}}DP( S_t) ≤ 4√2t. \]