Suppose that we are given an independent set \(I_0\) of a graph G, and an integer \(l\ge 0\). Then, we are asked to find an independent set of G having the maximum size among independent sets that are reachable from \(I_0\) by either adding or removing a single vertex at a time such that all intermediate independent sets are of size at least \(l\). We show that this problem is PSPACE-hard even for bounded-pathwidth graphs, and remains NP-hard for planar graphs. On the other hand, we give a linear-time algorithm to solve the problem for chordal graphs. We also study the parameterized complexity of the problem with respect to the following three parameters: the degeneracy \(d\) of an input graph, a lower bound \(l\) on the size of independent sets, and a lower bound \(s\) on the size of a solution reachable from \(I_0\). We show that the problem is fixed-parameter intractable when only one of \(d\), \(l\), or \(s\) is taken as a parameter. On the other hand, we give a fixed-parameter algorithm when parameterized by \(s+d\); this result implies that the problem parameterized only by \(s\) is fixed-parameter tractable for planar graphs, and for bounded-treewidth graphs.

假设我们得到了图G的独立集\（I_0 \）和一个整数\（l \ ge 0 \）。然后，要求我们通过一次添加或删除单个顶点，以使所有中间独立集的大小至少为一个值，从而从\（I_0 \）找到可从\（I_0 \）到达的独立集中具有最大大小的G的独立集。\（l \）。我们表明，即使对于有界路径图，这个问题也是PSPACE难题，而对于平面图，这个问题仍然是NP问题。另一方面，我们给出了线性时间算法来解决弦图的问题。我们还针对以下三个参数研究了问题的参数化复杂度：简并性输入图的\（d \），独立集的大小的下限\（l \），以及从\（I_0 \）可获得的解决方案的大小的下限\（s \）。我们表明，仅当\（d \），\（l \）或\（s \）中的一个作为参数时，问题是固定参数难以解决的。另一方面，当用\（s + d \）参数化时，我们给出了固定参数算法；该结果表明，仅由\（s \）参数化的问题对于平面图和有界树宽图是固定参数可处理的。