Partition of Sparse Multigraphs into a Forest and a Forest with Restrictions

The following measure of sparsity of multigraphs refining the maximum average degree: For $a>0$ and an arbitrary real $b$, a multigraph $H$ is \emph{$(a,b)$-sparse} if it is loopless and for every $A\subseteq V(H)$ with $|A|\geq 2$, the induced subgraph $H[A]$ has at most $a|A|+b$ edges. Forests are exactly $(1,-1)$-sparse multigraphs. It is known that the vertex set of any $(2,-1)$-sparse multigraph can be partitioned into two parts each of which induces a forest. For a given parameter $D$ we study for which pairs $(a,b)$ every $(a,b)$-sparse multigraph $G$ admits a vertex partition $(V_1, V_2)$ of $V(G)$ such that $G[V_1]$ and $G[V_2]$ are forests, and in addition either (i) $\Delta(G[V_1])\leq D$ or (ii) every component of $G[V_1]$ has at most $D$ edges. We find exact bounds on $a$ and $b$ for both types of problems (i) and (ii). We also consider problems of type (i) in the class of simple graphs and find exact bounds for all $D\geq 2$.