Independent Feedback Vertex Set Research Articles

Partitioning H-free graphs of bounded diameter

A natural way of increasing our understanding of NP-complete graph problems is to restrict the input to a special graph class. Classes of H-free graphs, that is, graphs that do not contain some graph H as an induced subgraph, have proven to be an ideal testbed for such a complexity study. However, if the forbidden graph H contains a cycle or claw, then these problems often stay NP-complete. A recent complexity study (MFCS 2019) on the k-Colouring problem shows that we may still obtain tractable results if we also bound the diameter of the H-free input graph. We continue this line of research by initiating a complexity study on the impact of bounding the diameter for a variety of classical vertex partitioning problems restricted to H-free graphs. We prove that bounding the diameter does not help for Independent Set, but leads to new tractable cases for problems closely related to 3-Colouring. That is, we show that Near-Bipartiteness, Independent Feedback Vertex Set, Independent Odd Cycle Transversal, Acyclic 3-Colouring and Star 3-Colouring are all polynomial-time solvable for chair-free graphs of bounded diameter. To obtain these results we exploit a new structural property of 3-colourable chair-free graphs.

Approximability of the independent feedback vertex set problem for bipartite graphs

Given an undirected graph G with n vertices, the independent feedback vertex set problem is to find a vertex subset F of G with the minimum number of vertices such that F is both an independent set and a feedback vertex set of G, if it exists. This problem is known to be NP-hard for bipartite planar graphs of maximum degree four. In this paper, we study the approximability of the problem. We first show that, for any fixed ε>0, unless P=NP, there exists no polynomial-time n1−ε-approximation algorithm even for bipartite planar graphs. We then give an α(Δ−1)/2-approximation algorithm for bipartite graphs G of maximum degree Δ, which runs in t(α,G)+O(Δn) time, under the assumption that there is an α-approximation algorithm for the original feedback vertex set problem on bipartite graphs which runs in t(α,G) time. This algorithmic result also yields a polynomial-time (exact) algorithm for the independent feedback vertex set problem on bipartite graphs of maximum degree three.

An Improved FPT Algorithm for Independent Feedback Vertex Set

We study the Independent Feedback Vertex Set problem — a variant of the classic Feedback Vertex Set problem where, given a graph G and an integer k, the problem is to decide whether there exists a vertex set Ssubseteq V(G) such that G ∖ S is a forest and S is an independent set of size at most k. We present an mathcal {O}^{ast }((1+varphi ^{2})^{k})-time FPT algorithm for this problem, where φ < 1.619 is the golden ratio, improving the previous fastest mathcal {O}^{ast }(4.1481^{k})-time algorithm given by Agrawal et al. (2016). The exponential factor in our time complexity bound matches the fastest deterministic FPT algorithm for the classic Feedback Vertex Set problem. On the technical side, the main novelty is a refined measure of an input instance in a branching process, that allows for a simpler and more concise description and analysis of the algorithm.

Recognition and Optimization Algorithms for P5-Free Graphs

The weighted independent set problem on P 5 -free graphs has numerous applications, including data mining and dispatching in railways. The recognition of P 5 -free graphs is executed in polynomial time. Many problems, such as chromatic number and dominating set, are NP-hard in the class of P 5 -free graphs. The size of a minimum independent feedback vertex set that belongs to a P 5 -free graph with n vertices can be computed in O ( n 16 ) time. The unweighted problems, clique and clique cover, are NP-complete and the independent set is polynomial. In this work, the P 5 -free graphs using the weak decomposition are characterized, as is the dominating clique, and they are given an O ( n ( n + m ) ) recognition algorithm. Additionally, we calculate directly the clique number and the chromatic number; determine in O ( n ) time, the size of a minimum independent feedback vertex set; and determine in O ( n + m ) time the number of stability, the dominating number and the minimum clique cover.

Independent Feedback Vertex Set for P_5-Free Graphs

The NP-complete problem Feedback Vertex Set is that of deciding whether or not it is possible, for a given integer kge 0, to delete at most k vertices from a given graph so that what remains is a forest. The variant in which the deleted vertices must form an independent set is called Independent Feedback Vertex Set and is also NP-complete. In fact, even deciding if an independent feedback vertex set exists is NP-complete and this problem is closely related to the 3-Colouring problem, or equivalently, to the problem of deciding whether or not a graph has an independent odd cycle transversal, that is, an independent set of vertices whose deletion makes the graph bipartite. We initiate a systematic study of the complexity of Independent Feedback Vertex Set for H-free graphs. We prove that it is NP-complete if H contains a claw or cycle. Tamura, Ito and Zhou proved that it is polynomial-time solvable for P_4-free graphs. We show that it remains polynomial-time solvable for P_5-free graphs. We prove analogous results for the Independent Odd Cycle Transversal problem, which asks whether or not a graph has an independent odd cycle transversal of size at most k for a given integer kge 0. Finally, in line with our underlying research aim, we compare the complexity of Independent Feedback Vertex Set for H-free graphs with the complexity of 3-Colouring, Independent Odd Cycle Transversal and other related problems.

Independent feedback vertex sets for graphs of bounded diameter

The Near-Bipartiteness problem is that of deciding whether or not the vertices of a graph can be partitioned into sets A and B, where A is an independent set and B induces a forest. The set A in such a partition is said to be an independent feedback vertex set. Yang and Yuan proved that Near-Bipartiteness is polynomial-time solvable for graphs of diameter 2 and NP-complete for graphs of diameter 4. We show that Near-Bipartiteness is NP-complete for graphs of diameter 3, resolving their open problem. We also generalise their result for diameter 2 by proving that even the problem of computing a minimum independent feedback vertex is polynomial-time solvable for graphs of diameter 2.

Algorithms for the Independent Feedback Vertex Set Problem

Algorithms for the Independent Feedback Vertex Set Problem

An improved parameterized algorithm for the independent feedback vertex set problem

In this paper, we develop a new parameterized algorithm for the Independent Feedback Vertex Set problem. Given a graph G=(V,E), the goal of the problem is to determine whether there exists a vertex subset F⊆V such that V−F induces a forest in G and F is an independent set. We show that there exists a parameterized algorithm that can determine whether a graph contains an IFVS of size k or not in time O(4kn2). To our best knowledge, this result improves the known upper bound for this problem, which is O(5knO(1)).

On Parameterized Independent Feedback Vertex Set

We investigate a generalization of the classical Feedback Vertex Set (FVS) problem from the point of view of parameterized algorithms. Independent Feedback Vertex Set (IFVS) is the “independent” variant of the FVS problem and is defined as follows: given a graph G and an integer k, decide whether there exists F⊆V(G), |F|≤k, such that G[V(G)∖F] is a forest and G[F] is an independent set; the parameter is k. Note that the similarly parameterized version of the FVS problem–where there is no restriction on the graph G[F]–has been extensively studied in the literature. The connected variant CFVS–where G[F] is required to be connected–has received some attention as well. The FVS problem easily reduces to the IFVS problem in a manner that preserves the solution size, and so any algorithmic result for IFVS directly carries over to FVS. We show that IFVS can be solved in O(5knO(1)) time, where n is the number of vertices in the input graph G, and obtain a cubic (O(k3)) kernel for the problem. Note the contrast with the CFVS problem, which does not admit a polynomial kernel unless CoNP⊆NP/Poly.

The NP-Completeness of Edge-Coloring

We show that it is NP-complete to determine the chromatic index of an arbitrary graph. The problem remains NP-complete even for cubic graphs.