Sagnik Sen - Academia.edu (original) (raw)

Papers by Sagnik Sen

Graph-Theoretic Concepts in Computer Science, 2021

For a class G of graphs, the problem Subgraph Complement to G asks whether one can find a subset ... more 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 K t (a complete graph on t vertices) for any fixed t ≥ 1, the problem is solvable in polynomial-time. This applies even when G is a subclass of K t-free graphs recognizable in polynomial-time, for example, the class of (t − 2)-degenerate graphs. • When H is a K 1,t (a star graph on t + 1 vertices), we obtain that the problem is NP-complete for every t ≥ 5. This, along with known results, leaves only two unresolved cases-K 1,3 and K 1,4. • When H is a P t (a path on t vertices), we obtain that the problem is NP-complete for every t ≥ 7, leaving behind only two unresolved cases-P 5 and P 6. • When H is a C t (a cycle on t vertices), we obtain that the problem is NP-complete for every t ≥ 8, leaving behind four unresolved cases-C 4 , C 5 , C 6 , and C 7. Further, we prove that these hard problems do not admit subexponential-time algorithms (algorithms running in time 2 o(|V (G)|)), assuming the Exponential Time Hypothesis. A simple complementation argument implies that results for G are applicable for G, thereby obtaining similar results for H being the complement of a complete graph, a star, a path, or a cycle. Our results generalize two main results and resolve one open question by Fomin et al. (Algorithmica, 2020).

Graph-Theoretic Concepts in Computer Science, 2021

For a class G of graphs, the problem Subgraph Complement to G asks whether one can find a subset ... more 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 K t (a complete graph on t vertices) for any fixed t ≥ 1, the problem is solvable in polynomial-time. This applies even when G is a subclass of K t-free graphs recognizable in polynomial-time, for example, the class of (t − 2)-degenerate graphs. • When H is a K 1,t (a star graph on t + 1 vertices), we obtain that the problem is NP-complete for every t ≥ 5. This, along with known results, leaves only two unresolved cases-K 1,3 and K 1,4. • When H is a P t (a path on t vertices), we obtain that the problem is NP-complete for every t ≥ 7, leaving behind only two unresolved cases-P 5 and P 6. • When H is a C t (a cycle on t vertices), we obtain that the problem is NP-complete for every t ≥ 8, leaving behind four unresolved cases-C 4 , C 5 , C 6 , and C 7. Further, we prove that these hard problems do not admit subexponential-time algorithms (algorithms running in time 2 o(|V (G)|)), assuming the Exponential Time Hypothesis. A simple complementation argument implies that results for G are applicable for G, thereby obtaining similar results for H being the complement of a complete graph, a star, a path, or a cycle. Our results generalize two main results and resolve one open question by Fomin et al. (Algorithmica, 2020).