1 Preliminary
The Feedback Vertex Set problem is one of the 21 problems proved to be NPhard by Karp [11]
. It asks to delete a minimum number of vertices to make a given graph into a forest. The problem has important applications in artificial intelligence
[1, 2], biocomputing [3, 10], operating system research [16], and so on.The Feedback Vertex Set problem has attracted a lot of attention from the parameterized complexity community. Both its undirected and directed versions are fixedparameter tractable when parameterized by the solution size [6, 7, 8, 9]. For the undirected case, the stateoftheart algorithm runs in time in the deterministic setting [6], and in the randomized setting [12]. Here the notation hides polynomial factors in , and is the solution size.
Relaxing the acyclic requirement, researchers have defined several classes of almost acyclic graphs. A graph is an pseudoforest if we can delete at most edges from each component in to get a forest. A pseudoforest is a 1pseudoforest. An almost forest is a graph from which we can delete edges to get a forest.
Philip et al.[14] introduced the problem of deleting vertices to get an almost acyclic graph. There are several results in this line of research. In [14], the authors gave an algorithm for pseudoforest deletion, which asks to delete at most vertices to get an pseudoforest. They also gave an time algorithm for the problem of pseudoforest deletion. Bodlaender et al. [4] gave an improved algorithm for pseudoforest deletion running in time .
Rai and Saurabh [15] gave an algorithm for the Almost Forest Deletion problem, which asks to delete at most vertices to get an almost forest. Lin et al.[13] gave an improved algorithm for this problem that runs in time .
To the author’s knowledge, there has been no followup research for the problem of pseudoforest deletion. In this paper, we provide a simple branching algorithm that runs in time . The in the algorithm by Philip et al. [14] is a large constant that depends on double exponentially. In contrast, the running time of our algorithm is independent of , which is somewhat surprising.
2 Notation and Terminology
Here we give a brief list of the graph theory concepts used in this paper; for other notation and terminology, we refer readers to [5].
For a graph , and are its vertex set and edge set respectively. A nonempty graph is connected if there is a path between every pair of vertices. Otherwise, we call it disconnected.
The multiplicity of an edge is the appearance number of it in the multigraph. An edge is a loop if , and we call it a loop at . The degree of a vertex is the number of its appearances as endvertex of some edge. Bypassing a vertex of degree 2 means to delete and add an edge between its two neighbors and (even if or there is already an edge between and ). A forest is a graph in which there is no cycle and a tree is a connected forest. A graph is a subgraph of a graph , if and . A subgraph of is an induced subgraph of if for every , edge if and only if . For , denotes the subgraph of induced by .
For a positive integer , a graph is an rpseudoforest if we can make each component into a forest by deleting at most edges.
3 FPT Algorithm for Pseudoforest Deletion
In this section, we give an FPT algorithm for pseudoforest deletion. Let us give the formal problem definition first.
Pseudoforest Deletion
Instance: An undirected graph , two integers and .
Parameter: .
Output: Decide if there is a set with such that is an pseudoforest.
Let be an instance of pseudoforest deletion, where contains edges and vertices.
Now we give some reduction rules to simplify the given instance.
Reduction Rule 1: If there is a component in that is an pseudoforest, then delete and get a new instance .
Reduction Rule 1 is safe since there is no need to delete any vertex from a component that is an pseudoforest.
Reduction Rule 2: If there are at least loops at a vertex , then delete and decrease by 1.
Reduction Rule 2 is safe since the graph is not an pseudoforest and every solution must contain .
Reduction Rule 3: If contains an edge of multiplicity greater than , then reduce its multiplicity to .
Every solution intersects if edge has multiplicity at least . Reducing the multiplicity of to does not affect the set of solutions. Thus, Reduction Rule 3 is safe.
Reduction Rule 4: If contains a vertex of degree at most 1, then delete and get a new instance .
On the one hand, every solution of is a solution of . On the other hand, attaching a leaf to any component of an pseudoforest still gives an pseudoforest. And so every solution of is also a solution of since . Thus, Reduction Rule 4 is safe.
Reduction Rule 5: If contains a vertex of degree 2, then bypass .
Lemma 1.
Reduction Rule 5 is safe.
Proof.
Let be the graph obtained from by bypassing . Let . We show that is a yesinstance if and only if is a yesinstance.
On the one hand, assume is a yesinstance, and is a solution of it. Suppose , then . By adding as an isolated vertex or a leaf to , we get . It follows that is an pseudoforest since is an pseudoforest. Thus, is also a solution of . Otherwise, . In this case, the edge is in some component of the pseudoforest . By subdividing in , we get , which is still an pseudoforest. And so is a solution of . In both cases, is a yesinstance, and is a solution of it.
On the other hand, assume is a yesinstance, and is a solution of it. Suppose . Consider . Note that . If , then . If , then . In both cases, is an pseudoforest. Thus is a solution for , and is a yesinstance.
Otherwise, . If , then we get from by bypassing . If , then we get from by deleting . In both cases, the number of edges and vertices decrease by the same amount, thus is also an pseudoforest. If , then we get from by deleting the isolated vertex . Above all, we know that is a yesinstance, and is a solution of it. ∎
Reduction Rule 6: If , terminate and conclude that is a noinstance.
We call a reduced instance if none of Reduction Rules 16 applies to it. Observe that if is a reduced instance, then satisfies the following conditions:

each edge has multiplicity at most ;

the minimum degree is at least 3;

there are at most loops at any vertex.
Lemma 2.
If a connected graph is an pseudoforest, then .
Proof.
We can make each connected component of an pseudoforest into a forest by deleting at most edges. Since is a connected pseudoforest, . ∎
Definition 1.
An pseudoforest with one vertex and loops is called an loop. An pseudoforest with two vertices and edges is called an edge.
Lemma 3.
Let be a reduced instance of pseudoforest deletion. Let be a subset of vertices such that is an pseudoforest. Suppose contains loops, components consists of one vertex and at least loops, edges, components consists of two vertices and at least edges. If , then the following statements hold.

.

.

For any , .
Proof.
Proof of statement 1: Because is an pseudoforest, we know that for each component in . Since the ratio decreases when increases, we have for each component in with at least 3 vertices. It follows that .
Proof of statement 2: Vertices in each edge have average degree and the vertex in an loop has degree . After exhaustive applications of Reduction Rule 1, there is no pseudoforest component in . Note that there are loops, components consists of one vertex and at least loops, edges, components consists of two vertices and at least edges in . Thus contains at least vertices of degree at least , vertices of degree at least and vertices of degree at least , vertices of degree at least . Combining with the fact that , we have
Let be an ordering of that satisfies . Let be the set of vertices with the largest degrees. The following lemma shows that every solution of a reduced instance intersects .
Lemma 4.
Let be any solution of pseudoforest deletion for a reduced instance , then .
Proof.
Suppose there is a solution of pseudoforest deletion on , which satisfies and . We show a contradiction by counting the number of edges in . Let be the resulting pseudoforest after deleting from . Suppose contains loops, components consisting of one vertex and at least loops, edges, and components consisting of two vertices and at least edges. Denote . In the following, we assume that , as otherwise, we may solve the problem via brute force. The assumption also implies that .
By the choice of , the degree of each vertex in is at most . Since , it follows that
(3) 
By the definition of and the assumption that , we have and so
(4) 
Combining inequalities (3) and (4), we have
(5) 
Since , it follows that
(6) 
Thus
According to Statement 2 in Lemma 3, we have
Therefore,
Set and simplify the above inequality, we get
It follows that , a contradiction to the fact that .
The above analysis shows that the assumption is not correct, thus every solution of must intersect with . ∎
Lemma 4 enables us to design the following algorithm for pseudoforest deletion.
Theorem 1.
The pseudoforest deletion problem parameterized by the solution size can be solved in time .
Proof.
Let be a given instance of the pseudoforest deletion problem, where is an undirected graph and is an integer. Each recursive call in Step 4 of Algorithm 1 decreases the parameter by 1, and thus the height of the search tree is at most . At each step, the problem branches into at most subproblems. Hence, the number of leaves in the search tree is at most . It follows that Algorithm 1 solves the problem of pseudoforest deletion and runs in time . ∎
4 Conclusion
In this paper, we design an FPT algorithm for the pseudoforest deletion problem. The running time of our algorithm is independent of , and so our algorithm improves over the result in [14], when is large compared with .
5 Acknowledgements
This research is supported by the National Natural Science Foundation of China (No. 61802178).
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