We continue the study of the recently-introduced C123-framework, for (simple) graph problems restricted to inputs specified by the forbidding of some finite set of subgraphs, to more general graph problems possibly involving multiedges and self-loops. We study specifically the problems Multigraph Matching Cut, Multigraph d-Cut and Partially Reflexive Stable Cut in this connection. The last may be seen as a Surjective Homomorphism problem to a path P_3 in which both leaves are looped while the interior vertex is loopless. We consider also another family of Surjective Homomorphism problems to a cycle in which only one vertex is loopless. When one forbids a single (simple) subgraph, our first three problems exhibit the same complexity behaviour as C123-problems, but on finite sets of forbidden subgraphs, the classification appears more complex. While Multigraph Matching Cut and Multigraph d-Cut have the same classification as C123-problems, already Partially Reflexive Stable Cut fails to have. This is witnessed by forbidding as subgraphs both C_3 and H_1. Indeed, the difference of behaviour occurs only around pendant subdivisions of nets and pendant subdivisions of H_1. We examine this area in close detail. Our other Surjective Homomorphism problem, ostensibly somewhat similar to Partially Reflexive Stable Cut, behaves very differently when the input is restricted to some class that is H-subgraph-free. For example, it is solvable in polynomial time on any class of bounded degree. Also, its hardness will never be preserved under any form of edge subdivision.

翻译：我们延续对近期提出的C123框架的研究，该框架最初针对（简单）图问题，其输入通过禁止有限子图集来定义，现将其扩展至可能涉及多重边和自环的更一般图问题。我们具体研究了与此相关的多重图匹配割、多重图d-割及部分自反稳定割问题。后者可视为到路径P_3的满射同态问题，其中两个端点带自环而内部顶点无自环。我们还研究了另一类到环的满射同态问题，其中仅一个顶点无自环。当禁止单个（简单）子图时，前三个问题展现出与C123问题相同的复杂性行为；但对于有限禁止子图集，分类模式显得更为复杂。虽然多重图匹配割和多重图d-割与C123问题具有相同的分类，但部分自反稳定割已出现差异。这一现象通过同时禁止C_3和H_1作为子图得到验证。事实上，行为差异仅出现在网结构的悬挂细分和H_1的悬挂细分附近。我们对此区域进行了细致考察。另一个满射同态问题表面上与部分自反稳定割相似，但在输入限制为某类H-子图自由图时表现出截然不同的性质：例如，在任何有界度图类上均存在多项式时间解法，且其计算困难性在任何形式的边细分操作下均不保持。