An Algorithm for the Feedback Vertex Set Problem on a Normal Helly Circular-Arc Graph

Hirotoshi Honma,Atsushi Sasaki,Yoko Nakajima

doi:10.4236/jcc.2016.48003

Abstract

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.

Highlights

A simple graph G is the intersection graph of if there exists a one-to-one correspondence between the vertices of G and the sets in, such that two vertices in G are adjacent if and only if their corresponding sets have a nonempty intersection
We propose an algorithm that takes O (n + m) time for the feedback vertex set (FVS) problem in a normal Helly circular-arc graph
We describe the procedure to construct an minimum cardinality FTS (MFTS) of normal Helly circular-arc graph (NHCG) G2 shown in Figure 3 by executing Step 3

Summary

Introduction

Let be a family of nonempty sets. A simple graph G is the intersection graph of if there exists a one-to-one correspondence between the vertices of G and the sets in , such that two vertices in G are adjacent if and only if their corresponding sets have a nonempty intersection. A graph G is called a circular-arc graph if it is the in-. (2016) An Algorithm for the Feedback Vertex Set Problem on a Normal Helly Circular-Arc Graph. Circular-arc graphs properly contain a class of interval graphs as a subclass. The FVS problem is known to be NP-hard for general graphs [15] and bipartite graphs [16]. Saha and Pal presented an algorithm that took O (n + m) time for the FVS problem in interval graphs using maximal clique decomposition [18]. Circular-arc graphs are a natural generalization of interval graphs. We propose an algorithm that takes O (n + m) time for the FVS problem in a normal Helly circular-arc graph.

Definitions and Notations

Algorithm and Its Correctness

Concluding Remarks