Decompositions of graphs with degree constraints relative to prescribed subgraphs

Given a finite simple undirected graph $G$, let $T_1(G)$ denote the subset of vertices of $G$ such that every vertex of $T_1(G)$ belongs to at least one subgraph isomorphic to a graph obtained by connecting a single vertex to two vertices of $K_4-e$. Define $T_0(G)=V(G)\setminus T_1(G)$, and let $a,b\text{\hspace{0.17em}⁣}:V(G)⟶{\mathbb{Z}}_{\ge 0}$ be arbitrary functions. In this paper, we prove that if $d_G(u)\ge a(u)+b(u)+h(u)$, where $h(u)\in \{0,1\}$ for $u\in T_h(G)$, then there exists a partition $(S,T)$ of $V(G)$ such that $d_S(u)\ge a(u)$ for every $u\in S$ and $d_T(u)\ge b(u)$ for every $u\in T$. This result extends the theorem of Stiebitz~[$\mathit{\text{J.}}\mathit{\text{Graph}}\mathit{\text{Theory}}$, 23 (1996), 321–324]. Moreover, we establish an analogous result in the case where $T_1(G)$ consists of vertices belonging to at least one $K_{2,3}$, thereby extending the findings of Hou et al.~[$\mathit{\text{Discrete}}\mathit{\text{Math.}}$, 341 (2018), 3288–3295].