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 $a,b \colon V(G) \longrightarrow \mathbb{Z}_{\ge 0}$ 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~[\textit{J. Graph 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.~[\textit{Discrete Math.}, 341 (2018), 3288--3295].