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 dₒ (u) a (u) for every u S and dₓ (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.