发表机构
Czech Technical University in Prague（布拉格捷克理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究分析了边约束图划分（ECGP）及其平衡、带符号变体的NP难性、多项式核、FPT算法与W[1]难性等参数化复杂性结果。
AI 中文摘要
我们研究边约束图划分问题（ECGP），该问题询问能否将图的顶点划分为r个部分，每个部分至少诱导出γ条边。我们还考虑了平衡变体（BECGP），要求各部分大小相等，以及带符号变体，其中一个部分的效用为其正边数与负边数之差。我们证明，对于固定的γ，ECGP和BECGP仍为NP难问题，而对于固定的r，BECGP也是NP难问题。对于自然参数r+γ，两个问题均存在多项式核。我们针对以最大叶子数、到团的顶点删除距离、簇顶点删除数加γ、顶点完整性为参数的ECGP和BECGP，获得了FPT算法。此外，ECGP在以到星型图的顶点删除距离加γ、到路径的顶点删除距离加γ为参数时，也是FPT算法。不利的是，当以r与若干结构参数共同为参数时，ECGP和BECGP是W[1]难问题。特别地，即使对应参数为零，该难解性仍适用于反馈边集、到星型图或路径的顶点删除距离、模宽。当以簇顶点删除数加r、团宽（即使γ=3）为参数时，问题也是W[1]难问题。对于带符号图，即使r+γ=3且输入是两个团的不交并，两个变体均为NP难问题。最后，平衡带符号变体在以树深度加r为参数时，即使γ=0，也是W[1]难问题。
英文摘要
We study the Edge-Constrained Graph Partitioning Problem (ECGP), which asks whether the vertices of a graph can be partitioned into r parts, each inducing at least gamma edges. We also consider a balanced variant (BECGP), requiring equal-sized parts, and signed variants, where the utility of a part is the difference between its numbers of positive and negative edges. We show that ECGP and BECGP remain NP-hard for fixed gamma, while BECGP is also NP-hard for fixed r. For the natural parameterization r+gamma, both problems admit polynomial kernels. We obtain FPT algorithms for ECGP and BECGP parameterized by maximum leaf number, vertex deletion distance to a clique, cluster vertex deletion number plus gamma, and vertex integrity. Furthermore, ECGP is FPT parameterized by vertex deletion distance to stars plus gamma and vertex deletion distance to paths plus gamma. On the negative side, ECGP and BECGP are W[1]-hard when parameterized by r together with several structural parameters. In particular, hardness holds for feedback edge set, vertex deletion distance to stars or paths, and modular width even when the corresponding parameter is zero. The problems are also W[1]-hard parameterized by cluster vertex deletion number plus r, and by clique-width even when gamma=3. For signed graphs, both variants are NP-hard even when r+gamma=3 and the input is a disjoint union of two cliques. Finally, the balanced signed variant is W[1]-hard parameterized by treedepth plus r, even when gamma=0.
