Graph-modification problems, where we add/delete a small number of vertices/edges to make the given graph to belong to a simpler graph class, is a well-studied optimization problem in all algorithmic paradigms including classical, approximation and parameterized complexity. Specifically, graph-deletion problems, where one needs to delete at most $k$ vertices to place it in a given non-trivial hereditary (closed under induced subgraphs) graph class, captures several well-studied problems including {\sc Vertex Cover}, {\sc Feedback Vertex Set}, {\sc Odd Cycle Transveral}, {\sc Cluster Vertex Deletion}, and {\sc Perfect Deletion}. Investigation into these problems in parameterized complexity has given rise to powerful tools and techniques. While a precise characterization of the graph classes for which the problem is {\it fixed-parameter tractable} (FPT) is elusive, it has long been known that if the graph class is characterized by a {\it finite} set of forbidden graphs, then the problem is FPT. In this paper, we initiate a study of a natural variation of the problem of deletion to {\it scattered graph classes} where we need to delete at most $k$ vertices so that in the resulting graph, each connected component belongs to one of a constant number of graph classes. A simple hitting set based approach is no longer feasible even if each of the graph classes is characterized by finite forbidden sets. As our main result, we show that this problem is fixed-parameter tractable (FPT) when the deletion problem corresponding to each of the finite classes is known to be FPT and the properties that a graph belongs to each of the classes is expressible in CMSO logic. When each graph class has a finite forbidden set, we give a faster FPT algorithm using the well-known techniques of iterative compression and important separators.
翻译:图形修改问题, 我们在此添加/ 删除少量的顶点/ 边缘, 使给定的图表属于更简单的图形类, 是所有算法范式( 包括古典、 近似和参数化复杂性) 中一个研究周密的优化问题。 具体地说, 图形删除问题, 需要以最多 $k$ 的顶点删除, 才能将其置于一个非三角性遗传性( 以诱导的子图) 图表类中, 捕捉到一些研究周密的图形类问题, 包括 ~sc Vertex Coverlate}, ~sc 反馈甚快的Vertex Set}, ~c Odclorver Transver}, ~ croc Croad Vertex dedeltion }, 以及~ sexcreal commal rial rial mailations dislates the liver the liver the liver listal lives the we fral fral fral fral frm le) ma is the we knal fol lest le isn lest the we mold fral fral fral fral frmst mal lest fral frmlem frm frm frm) 。 虽然 。 当我们所知道 the frm frm frm frm frm frm frm frmisal is frmis 发现我们所知道时, 我们是知道, 问题是知道, 我们是知道的每个的每个的每个的每在使用的每个硬性研究, 我们所知道 问题是知道, 问题是已知的每个硬性的每个硬性, 问题, 问题是已知, 问题是已知, 问题是已知的每在使用直。我们所知道, 问题是已知的每个硬性地的每个硬性地研究中, 然后我们所知道, 问题。