Let π be a family of graphs. In the classical π-Vertex Deletion problem, given a graph G and a positive integer k, the objective is to check whether there exists a subset S of at most k vertices such that G − S is in π. In this paper, we introduce the conflict free version of this classical problem, namely Conflict Free π-Vertex Deletion (CF-π-VD), and study this problem from the viewpoint of classical and parameterized complexity. In the CF-π-VD problem, given two graphs G and H on the same vertex set and a positive integer k, the objective is to determine whether there exists a set \(S\subseteq V(G)\), of size at most k, such that G − S is in π and H[S] is edgeless. Initiating a systematic study of these problems is one of the main conceptual contribution of this work. We obtain several results on the conflict free versions of several classical problems. Our first result shows that if π is characterized by a finite family of forbidden induced subgraphs then CF-π-VD is Fixed Parameter Tractable (FPT). Furthermore, we obtain improved algorithms for conflict free versions of several well studied problems. Next, we show that if π is characterized by a “well-behaved” infinite family of forbidden induced subgraphs, then CF-π-VD is W[1]-hard. Motivated by this hardness result, we consider the parameterized complexity of CF-π-VD when H is restricted to well studied families of graphs. In particular, we show that the conflict free version of several well-known problems such as Feedback Vertex Set, Odd Cycle Transversal, Chordal Vertex Deletion and Interval Vertex Deletion are FPT when H belongs to the families of d-degenerate graphs and nowhere dense graphs.

中文翻译：

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图上覆盖问题的无冲突版本：经典和参数化

令π为图族。在经典的π-顶点删除的问题，给定图ģ和一个正整数ķ，目的是检查是否存在一个子集小号至多的ķ顶点使得ģ -小号是π。在本文中，我们介绍了该经典问题的无冲突版本，即无冲突π顶点删除（CF-π-VD），并从经典和参数化复杂性的角度研究了该问题。在CF-π-VD问题中，在相同的顶点集上给定两个图G和H和一个正整数ķ，目标是，以确定是否存在一组\（S \ subseteq V（G）\）至多，大小的ķ，使得G ^ -小号是在π和ħ [小号]是无边界。对这些问题进行系统的研究是这项工作的主要概念贡献之一。我们获得了几个经典问题的无冲突版本的几个结果。我们的第一个结果表明，如果π的特征是有限的禁止诱导子图族，则CF-π-VD是固定参数可牵引的（FPT）。此外，我们获得了一些经过深入研究的问题的无冲突版本的改进算法。接下来，我们表明，如果π的特征是“行为良好”的无穷禁止子图族，则CF-π-VD为W [1] -hard。受此硬度结果的影响，当H限于经过充分研究的图形族时，我们考虑CF-π-VD的参数化复杂性。特别地，我们证明了当H时，一些众所周知的问题（如反馈顶点集，奇数周期遍历，弦顶点删除和间隔顶点删除）的无冲突版本是FPT。属于d-退化图族，而无处密集图。