Decompositions of graphs with degree constraints relative to prescribed subgraphs

Given a finite simple undirected graph G, let T₁(G) denote the subset of vertices of G such that every vertex of T₁(G) belongs to at least one subgraph isomorphic to a graph obtained by connecting a single vertex to two vertices of K₄ - e. Define T₀(G) = V(G) T₁(G), and let a,b V(G) Z≥ 0 be arbitrary functions. In this paper, we prove that if dG(u) ≥ a(u) + b(u) + h(u), where h(u) ∈ \0,1\ for u ∈ Tₕ(G), then there exists a partition $(S, T)$ of $V(G)$ such that dS(u) ≥ a(u) for every u ∈ S and dT(u) ≥ b(u) for every u ∈ T. This result extends the theorem of Stiebitz~[J. Graph Theory, 23 (1996), 321--324]. Moreover, we establish an analogous result in the case where T₁(G) consists of vertices belonging to at least one K2,3, thereby extending the findings of Hou et al.~[Discrete Math., 341 (2018), 3288--3295].