# Node-and edge-deletion NP-complete problems

@article{Yannakakis1978NodeandEN, title={Node-and edge-deletion NP-complete problems}, author={Mihalis Yannakakis}, journal={Proceedings of the tenth annual ACM symposium on Theory of computing}, year={1978} }

If &pgr; is a graph property, the general node(edge) deletion problem can be stated as follows: Find the minimum number of nodes(edges), whose deletion results in a subgraph satisfying property &pgr;. In this paper we show that if &pgr; belongs to a rather broad class of properties (the class of properties that are hereditary on induced subgraphs) then the node-deletion problem is NP-complete, and the same is true for several restrictions of it. For the same class of properties, requiring the…

## 471 Citations

### Edge-Deletion Problems

- Mathematics, Computer ScienceSIAM J. Comput.
- 1981

This paper shows that the edge-deletion problem is NP-complete for the following properties: (1) without cycles of specified length l, or of any length $ \leqq l$, (2) connected and degree-constrained, (3) outerplanar, (4) transitive digraph, (5) line-invertible, (6) bipartite, (7)transitively orientable.

### Edge-deletion and edge-contraction problems

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- 1982

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- 1995

NC algorithms for solving the problem of finding a maximal subset F of edges in G such that the subgraph induced by F is acyclic are presented and it is shown that the problem is solvable in NC if the input graph G has only vertex-induced paths of length polylogarithmic in the number of vertices of G.

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### Edge Deletion to Restrict the Size of an Epidemic

- Mathematics, Computer ScienceArXiv
- 2021

It is proved that the Th+1-Free Edge Deletion problem is fixed-parameter tractable (FPT) when parameterized by the vertex cover number and that it admits a kernel with O(hk) vertices and O( hk) edges, when parameterizing by combined parameters h and the solution size k.

### Strong Parameterized Deletion: Bipartite Graphs

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This paper studies Strong Bipartite Deletion, where given an undirected graph G and positive integers k and l, the objective is to check whether there exists a vertex subset S of size at most k such that each connected component of G-S can be made bipartite by deleting at most l edges.

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- Mathematics, Computer ScienceArXiv
- 2018

It is proved that the FairVC problem is W[1]-hard with parameterization by both treedepth and feedback vertex set of the input graph and an FPT algorithm is provided for the Fair Vertex Cover problem parameterized by modular width.

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