On subgraph complementation to H-free graphs

@inproceedings{Antony2021OnSC,
  title={On subgraph complementation to H-free graphs},
  author={Dhanyamol Antony and Jay Garchar and Sagartanu Pal and R. B. Sandeep and Sagnik Sen and R. Subashini},
  booktitle={WG},
  year={2021}
}
For a class G of graphs, the problem Subgraph Complement to G asks whether one can find a subset S of vertices of the input graph G such that complementing the subgraph induced by S in G results in a graph in G. We investigate the complexity of the problem when G is H-free for H being a complete graph, a star, a path, or a cycle. We obtain the following results: • When H is a Kt (a complete graph on t vertices) for any fixed t ≥ 1, the problem is solvable in polynomial-time. This applies even… 

Cutting a tree with Subgraph Complementation is hard, except for some small trees

For a graph property Π , Subgraph Complementation to Π is the problem to find whether there is a subset S of vertices of the input graph G such that modifying G by complementing the subgraph induced

References

SHOWING 1-10 OF 19 REFERENCES

Incompressibility of H-free edge modification problems: Towards a dichotomy

TLDR
A set of nine 5-vertex graphs whose incompressibility would give a complete classification of the kernelization complexity of H-free Edge Editing for every graph $H$ with at least 5 vertices that is neither complete nor empty.

Polynomial Kernels for Paw-free Edge Modification Problems

TLDR
This work answers both questions affirmatively by presenting, respectively, $O (k)$-vertex and $O(k^4)$ -vertex kernels for them as part of an ongoing program that aims at understanding compressibility of H-free edge modification problems.

Subgraph Complementation

TLDR
It is shown that this problem can be solved in polynomial time for various choices of the graphs class G, such as bipartite, d -degenerate, or cographs, and that the problem is NP -complete when G is the class of regular graphs.

A Polynomial Kernel for Paw-Free Editing

TLDR
The question of compressibility for one of the last two unresolved graphs H on $4$ vertices is positively answered and the first polynomial kernel for paw-free editing with $O(k^{6})$vertices is given.

Which Problems Have Strongly Exponential Complexity

TLDR
A generalized reduction that is based on an algorithm that represents an arbitrary k-CNF formula as a disjunction of 2?nk-C NF formulas that are sparse, that is, each disjunct has O(n) clauses, and shows that Circuit-SAT is SERF-complete for all NP-search problems.

Edge-Deletion Problems

TLDR
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.

Dichotomy Results on the Hardness of H-free Edge Modification Problems

TLDR
It is proved that these NP-complete problems cannot be solved in parameterized subexponential time, i.e., in time $2^{o(k)}\cd...

On Switching to H‐Free Graphs

TLDR
The problem of deciding if, for a fixed graph H, a given graph is switching equivalent to an H-free graph is studied, showing that for H isomorphic to a claw, the problem is polynomial, and giving infinitely many graphs H such that the problems are NP-complete.

Parameterized Algorithms

TLDR
This comprehensive textbook presents a clean and coherent account of most fundamental tools and techniques in Parameterized Algorithms and is a self-contained guide to the area, providing a toolbox of algorithmic techniques.

Minimum Fill-In: Inapproximability and Almost Tight Lower Bounds

TLDR
This paper presents a very weak approximation algorithm based on ETH, and a strong one based on hardness of subexponential-time approximation of the minimum bisection problem on regular graphs, both of which are strong approximation schemes for any positive $\delta$, assuming ETH.