In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
discipline of
graph theory
In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
, a feedback vertex set (FVS) of a
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
is a set of vertices whose removal leaves a graph without
cycles ("removal" means deleting the vertex and all edges adjacent to it). Equivalently, each FVS contains at least one vertex of any cycle in the graph. The feedback vertex set number of a graph is the size of a smallest feedback vertex set. The minimum feedback vertex set problem is an
NP-complete
In computational complexity theory, a problem is NP-complete when:
# it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying ...
problem; it was among the
first problems shown to be NP-complete. It has wide applications in
operating system
An operating system (OS) is system software that manages computer hardware, software resources, and provides common daemon (computing), services for computer programs.
Time-sharing operating systems scheduler (computing), schedule tasks for ef ...
s,
database system
In computing, a database is an organized collection of data stored and accessed electronically. Small databases can be stored on a file system, while large databases are hosted on computer clusters or cloud storage. The design of databases spa ...
s, and
VLSI chip design.
Definition
The FVS
decision problem
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question of the input values. An example of a decision problem is deciding by means of an algorithm whe ...
is as follows:
:INSTANCE: An (undirected or directed)
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
and a positive integer
.
:QUESTION: Is there a subset
with
such that, when all vertices of
and their adjacent edges are deleted from
, the remainder is
cycle-free?
The graph