Connected Feedback Vertex Set Research Articles

On Cycle Transversals and Their Connected Variants in the Absence of a Small Linear Forest

A graph is H-free if it contains no induced subgraph isomorphic to H.We prove new complexity results for the two classical cycle transversal problems Feedback Vertex Set and Odd Cycle Transversal by showing that they can be solved in polynomial time on (sP_1+ P_3)-free graphs for every integer sge 1.We show the same result for the variants Connected Feedback Vertex Set and Connected Odd Cycle Transversal. We also prove that the latter two problems are polynomial-time solvable on cographs; this was already known for Feedback Vertex Set and Odd Cycle Transversal. We complement these results by proving that Odd Cycle Transversal and Connected Odd Cycle Transversal are NP-complete on (P_2+ P_5,P_6)-free graphs.

Minimum connected transversals in graphs: New hardness results and tractable cases using the price of connectivity

We perform a systematic study in the computational complexity of the connected variant of three related transversal problems: Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. Just like their original counterparts, these variants are NP-complete for general graphs. A graph G is H-free for some graph H if G contains no induced subgraph isomorphic to H. It is known that Connected Vertex Cover is NP-complete even for H-free graphs if H contains a claw or a cycle. We show that the two other connected variants also remain NP-complete if H contains a cycle or claw. In the remaining case H is a linear forest. We show that Connected Vertex Cover, Connected Feedback Vertex Set, and Connected Odd Cycle Transversal are polynomial-time solvable for sP2-free graphs for every constant s≥1. For proving these results we use known results on the price of connectivity for vertex cover, feedback vertex set, and odd cycle transversal. This is the first application of the price of connectivity that results in polynomial-time algorithms.

Dynamic Parameterized Problems

We study the parameterized complexity of various graph theoretic problems in the dynamic framework where the input graph is being updated by a sequence of edge additions and deletions. Vertex subset problems on graphs typically deal with finding a subset of vertices having certain properties. In real world applications, the graph under consideration often changes over time and due to this dynamics, the solution at hand might lose the desired properties. The goal in the area of dynamic graph algorithms is to efficiently maintain a solution under these changes. Recomputing a new solution on the new graph is an expensive task especially when the number of modifications made to the graph is significantly smaller than the size of the graph. In the context of parameterized algorithms, two natural parameters are the size k of the symmetric difference of the edge sets of the two graphs (on n vertices) and the size r of the symmetric difference of the two solutions. We study the Dynamic $$\Pi $$ -Deletion problem which is the dynamic variant of the classical $$\Pi $$ -Deletion problem and show NP-hardness, fixed-parameter tractability and kernelization results. For specific cases of Dynamic $$\Pi $$ -Deletion such as Dynamic Vertex Cover and Dynamic Feedback Vertex Set, we describe improved algorithms and linear kernels. Specifically, we show that Dynamic Vertex Cover has a deterministic algorithm with $$1.0822^k n^{\mathcal {O}(1)}$$ running time and Dynamic Feedback Vertex Set has a randomized algorithm with $$1.6667^k n^{\mathcal {O}(1)}$$ running time. We also show that Dynamic Connected Feedback Vertex Set can be solved in $$2^{\mathcal {O}(k)} n^{\mathcal {O}(1)}$$ time. For each of Dynamic Connected Vertex Cover, Dynamic Dominating Set and Dynamic Connected Dominating Set, we describe an algorithm with $$2^k n^{\mathcal {O}(1)}$$ running time and show that this is the optimal running time (up to polynomial factors) assuming the Set Cover Conjecture.

The price of connectivity for feedback vertex set

Let fvs(G) and cfvs(G) denote the cardinalities of a minimum feedback vertex set and a minimum connected feedback vertex set of a graph G, respectively. The price of connectivity for feedback vertex set (poc-fvs) for a class of graphs G is defined as the maximum ratio cfvs(G)/fvs(G) over all connected graphs G∈G. We study the poc-fvs for graph classes defined by a finite family H of forbidden induced subgraphs. We characterize exactly those finite families H for which the poc-fvs for H-free graphs is upper bounded by a constant. Additionally, for the case where ∣H∣=1, we determine exactly those graphs H for which there exists a constant cH such that cfvs(G)≤fvs(G)+cH for every connected H-free graph G, as well as exactly those graphs H for which we can take cH=0.

Kernelization hardness of connectivity problems in [formula omitted]-degenerate graphs

A graph is d-degenerate if every subgraph contains a vertex of degree at most d. For instance, planar graphs are 5-degenerate. Inspired by the recent work by Philip, Raman and Sikdar, who have shown the existence of a polynomial kernel for Dominating Set in d-degenerate graphs, we investigate the kernelization complexity of problems that include a connectivity requirement in this class of graphs.Our main contribution is the proof that Connected Dominating Set does not admit a polynomial kernel in d-degenerate graphs for d≥2 unless NP⊆coNP/poly, which is known to cause a collapse of the polynomial hierarchy to its third level. We prove this using a problem that originates from bioinformatics–Colourful Graph Motif–analysed and proved to be NP-hard by Fellows et al. This problem nicely encapsulates the hardness of the connectivity requirement in kernelization. Our technique also yields an alternative proof that, under the same complexity assumption, Steiner Tree does not admit a polynomial kernel. The original proof, via a reduction from Set Cover, is due to Dom, Lokshtanov and Saurabh.We extend our analysis by showing that, unless NP⊆coNP/poly, there do not exist polynomial kernels for Steiner Tree, Connected Feedback Vertex Set and Connected Odd Cycle Transversal in d-degenerate graphs for d≥2. On the other hand, we show a polynomial kernel for Connected Vertex Cover in graphs that do not contain the biclique Ki,j as a subgraph. As a d-degenerate graph cannot contain Kd+1,d+1 as a subgraph, the results holds also for graphs of bounded degeneracy.

FPT algorithms for Connected Feedback Vertex Set

We study the recently introduced Connected Feedback Vertex Set (CFVS) problem from the view-point of parameterized algorithms. CFVS is the connected variant of the classical Feedback Vertex Set problem and is defined as follows: given a graph G=(V,E) and an integer k, decide whether there exists F?V, |F|?k, such that G[V?F] is a forest and G[F] is connected. We show that Connected Feedback Vertex Set can be solved in time O(2 O(k) n O(1)) on general graphs and in time $O(2^{O(\sqrt{k}\log k)}n^{O(1)})$ on graphs excluding a fixed graph H as a minor. Our result on general undirected graphs uses, as a subroutine, a parameterized algorithm for Group Steiner Tree, a well studied variant of Steiner Tree. We find the algorithm for Group Steiner Tree of independent interest and believe that it could be useful for obtaining parameterized algorithms for other connectivity problems.