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The feedback vertex set (FVS) problem is to find the set of vertices of minimum cardinality whose removal renders the graph acyclic. The FVS problem has applications in several areas such as combinatorial circuit design, synchronous systems, computer systems, and very-large-scale integration (VLSI) circuits. The FVS problem is known to be NP-hard for simple graphs, but polynomi-al-time algorithms have been found for special classes of graphs. The intersection graph of a collection of arcs on a circle is called a circular-arc graph. A normal Helly circular-arc graph is a proper subclass of the set of circular-arc graphs. In this paper, we present an algorithm that takes time to solve the FVS problem in a normal Helly circular-arc graph with n vertices and m edges.

Let

Let

The FVS problem is known to be NP-hard for general graphs [

The remainder of this paper is organized as follows. We state the definitions and notations used throughout this paper in Section 2. Next, we present our algorithm for the FVS problem and analyze its complexity in Section 3. Finally, we summarize our findings in Section 4 and conclude the paper by briefly discussing the scope for future work.

In this section, we provide the definitions and relevant notations used throughout the paper. These establish the basis of the algorithm presented in Section 3. We provide the definitions of a circular-arc model and its corresponding graph. Consider a unit circle C and a family

Normal and Helly circular-arc models (NHCM) are precisely those without three or less arcs covering the entire circle [

A maximal clique is a clique to which no further vertices of a graph can be added such that it remains a clique. For the graph

Let r be the number of maximal cliques of NHCG G. Throughout this paper, we use the term triangle to denote a cycle whose length is three. We define functions

Thus,

A chordal graph is a simple graph in which every cycle of length four or greater has a cycle chord. Interval graphs are a subclass of chordal graphs [

In this section, we present an algorithm for solving the FVS problem for an NHCG. We will concisely describe the outline of our algorithm. First, we decompose a given NHCG into maximal cliques. An FTS is obtained by removing

Let

We use the graph

Begin

(Step 1)

(Step 2)

1st iteration

(1-1)

(1-2)

(1-3)

2nd iteration

(2-1)

(2-2)

3rd iteration

(3-1)

(3-2)

(Step 3)

End

In Step 1, all maximal cliques can be generated in

Lemma 1. Let G be an NHCG. Following the execution of Step 2 of Algorithm 1, F is an MFTS of G.

Proof: Each triangle contained in G is a subset of any maximal clique

In Step 2, initially, we set

In (1-1) of Step 2, we select all vertices except two minima with

In (1-2) of Step 2, if

Similarly, in the next step (1-3), if

In the second iteration, we update

Here, we explain how

In Step 2, we select

Next, we consider the case of a maximal clique

Thus far, we have presented an example where an MFTS of an NHCG G is also its MFVS. However, there exist cases where an MFTS of G obtained by executing Step 2 of Algorithm 1 is not an MFVS of G. We describe the procedure to construct an MFTS of NHCG

Begin

(Step 1)

(Step 2)

1st iteration

(1-1)

(1-2)

(1-3)

2nd iteration

(2-1)

(2-2)

(2-3)

3rd iteration

(3-1)

(3-2)

(Step 3)

We have

End

Following the execution of Step 2 of Algorithm 1, we obtain an MFTS

The following lemmas guarantee the validity of Algorithm 1.

Lemma 2. Let G be a normal Helly circular-arc graph. If F is an MFTS and not an MFVS of G, a periphery in

Proof: As mentioned in Section 2, interval graphs are a subclass of NHCGs and have no periphery. An NHCM from which all back-arcs are removed is topologically equivalent to an interval model. Therefore,

Thus, if F is an MFTS and not an MFVS of

Lemma 3. Let G be a normal Helly circular-arc graph. Following the execution of Step 3 of Algorithm 1, F is an MFVS of G.

Proof: By Lemma 2, if

In the case where

We consider the cases where

two possible cases where

Therefore, we can construct an MFVS after executing Step 3 of Algorithm 1.

In the following, we analyze the complexity of Algorithm 1. In Step 1, all maximal cliques of G are computed in

Theorem 1. Algorithm 1 finds an MFVS of a normal Helly circular-arc graph G in

In this paper, we proposed an algorithm that takes

We thank the Editor and the referee for their comments. This work was partially supported by JSPS KAKENHI Grant Number 25330019.

Hirotoshi Honma,Yoko Nakajima,Atsushi Sasaki, (2016) An Algorithm for the Feedback Vertex Set Problem on a Normal Helly Circular-Arc Graph. Journal of Computer and Communications,04,23-31. doi: 10.4236/jcc.2016.48003