In graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a set of vertices such that for every two vertices in , there is no edge connecting the two. Equivalently, each edge in the graph has at most one endpoint in . The size of an independent set is the number of vertices it contains. Independent sets have also been called internally stable sets.
A maximal independent set is an independent set that is not a proper subset of any other independent set.
A maximum independent set is an independent set of largest possible size for a given graph . This size is called the independence number of , and denoted . The problem of finding such a set is called the maximum independent set problem and is an NP-hard optimization problem. As such, it is unlikely that there exists an efficient algorithm for finding a maximum independent set of a graph.
Every maximum independent set also is maximal, but the converse implication does not necessarily hold.
A set is independent if and only if it is a clique in the graph’s complement, so the two concepts are complementary. In fact, sufficiently large graphs with no large cliques have large independent sets, a theme that is explored in Ramsey theory.
A set is independent if and only if its complement is a vertex cover. and the size of a minimum vertex cover is equal to the number of vertices in the graph.Therefore, the sum of the size of the largest independent set
A vertex coloring of a graph corresponds to a partition of its vertex set into independent subsets. Hence the minimal number of colors needed in a vertex coloring, the chromatic number, is at least the quotient of the number of vertices in and the independent number .
In a bipartite graph with no isolated vertices, the number of vertices in a maximum independent set equals the number of edges in a minimum edge covering; this is Kőnig's theorem.
An independent set that is not a proper subset of another independent set is called maximal. Such sets are dominating sets. Every graph contains at most 3n/3 maximal independent sets,but many graphs have far fewer. The number of maximal independent sets in n-vertex cycle graphs is given by the Perrin numbers, and the number of maximal independent sets in n-vertex path graphs is given by the Padovan sequence. Therefore, both numbers are proportional to powers of 1.324718..., the plastic number.
In computer science, several computational problems related to independent sets have been studied.
The first three of these problems are all important in practical applications; the independent set decision problem is not, but is necessary in order to apply the theory of NP-completeness to problems related to independent sets.
The independent set problem and the clique problem are complementary: a clique in G is an independent set in the complement graph of G and vice versa. Therefore, many computational results may be applied equally well to either problem. For example, the results related to the clique problem have the following corollaries:
Despite the close relationship between maximum cliques and maximum independent sets in arbitrary graphs, the independent set and clique problems may be very different when restricted to special classes of graphs. For instance, for sparse graphs (graphs in which the number of edges is at most a constant times the number of vertices in any subgraph), the maximum clique has bounded size and may be found exactly in linear time;however, for the same classes of graphs, or even for the more restricted class of bounded degree graphs, finding the maximum independent set is MAXSNP-complete, implying that, for some constant c (depending on the degree) it is NP-hard to find an approximate solution that comes within a factor of c of the optimum.
The maximum independent set problem is NP-hard. However, it can be solved more efficiently than the O(n2 2n) time that would be given by a naive brute force algorithm that examines every vertex subset and checks whether it is an independent set.
As of 2017 it can be solved in time O(1.1996n) using polynomial space.When restricted to graphs with maximum degree 3, it can be solved in time O(1.0836n).
For many classes of graphs, a maximum weight independent set may be found in polynomial time. Famous examples are claw-free graphs,P5-free graphs and perfect graphs. For chordal graphs, a maximum weight independent set can be found in linear time.
Modular decomposition is a good tool for solving the maximum weight independent set problem; the linear time algorithm on cographs is the basic example for that. Another important tool are clique separators as described by Tarjan.
Kőnig's theorem implies that in a bipartite graph the maximum independent set can be found in polynomial time using a bipartite matching algorithm.
In general, the maximum independent set problem cannot be approximated to a constant factor in polynomial time (unless P = NP). In fact, Max Independent Set in general is Poly-APX-complete, meaning it is as hard as any problem that can be approximated to a polynomial factor.However, there are efficient approximation algorithms for restricted classes of graphs.
In planar graphs, the maximum independent set may be approximated to within any approximation ratio c < 1 in polynomial time; similar polynomial-time approximation schemes exist in any family of graphs closed under taking minors.
In bounded degree graphs, effective approximation algorithms are known with approximation ratios that are constant for a fixed value of the maximum degree; for instance, a greedy algorithm that forms a maximal independent set by, at each step, choosing the minimum degree vertex in the graph and removing its neighbors, achieves an approximation ratio of (Δ+2)/3 on graphs with maximum degree Δ. Approximation hardness bounds for such instances were proven in Berman & Karpinski (1999). Indeed, even Max Independent Set on 3-regular 3-edge-colorable graphs is APX-complete.
An interval graph is a graph in which the nodes are 1-dimensional intervals (e.g. time intervals) and there is an edge between two intervals iff they intersect. An independent set in an interval graph is just a set of non-overlapping intervals. The problem of finding maximum independent sets in interval graphs has been studied, for example, in the context of job scheduling: given a set of jobs that has to be executed on a computer, find a maximum set of jobs that can be executed without interfering with each other. This problem can be solved exactly in polynomial time using earliest deadline first scheduling.
A geometric intersection graph is a graph in which the nodes are geometric shapes and there is an edge between two shapes iff they intersect. An independent set in a geometric intersection graph is just a set of disjoint (non-overlapping) shapes. The problem of finding maximum independent sets in geometric intersection graphs has been studied, for example, in the context of Automatic label placement: given a set of locations in a map, find a maximum set of disjoint rectangular labels near these locations.
Finding a maximum independent set in intersection graphs is still NP-complete, but it is easier to approximate than the general maximum independent set problem. A recent survey can be found in the introduction of Chan & Har-Peled (2012).
The problem of finding a maximal independent set can be solved in polynomial time by a trivial greedy algorithm.All maximal independent sets can be found in time O(3n/3) = O(1.4423n).
|Name||License||API language||Brief info|
|igraph||GPL||C, Python, R, Ruby||exact solution. "The current implementation was ported to igraph from the Very Nauty Graph Library by Keith Briggs and uses the algorithm from the paper S. Tsukiyama, M. Ide, H. Ariyoshi and I. Shirawaka. A new algorithm for generating all the maximal independent sets. SIAM J Computing, 6:505–517, 1977".|
|LightGraphs||MIT||Julia||exact solution. See the documentation for more details.|
|NetworkX||BSD||Python||approximate solution, see the routine maximum_independent_set|
|mis||Open||Rust (binaries)||approximate solution by random sampling of maximal independent set space, see web page for more details|
The maximum independent set and its dual, the minimum vertex cover problem, is involved in proving the computational complexity of many theoretical problems.They also serve as useful models for real world optimization problems, for example minimum independent set is a useful model for discovering stable genetic components for designing engineered genetic systems.
In computer science, the clique problem is the computational problem of finding cliques in a graph. It has several different formulations depending on which cliques, and what information about the cliques, should be found. Common formulations of the clique problem include finding a maximum clique, finding a maximum weight clique in a weighted graph, listing all maximal cliques, and solving the decision problem of testing whether a graph contains a clique larger than a given size.
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color.
In the mathematical area of graph theory, a clique is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent; that is, its induced subgraph is complete. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs. Cliques have also been studied in computer science: the task of finding whether there is a clique of a given size in a graph is NP-complete, but despite this hardness result, many algorithms for finding cliques have been studied.
In the mathematical discipline of graph theory, a vertex cover of a graph is a set of vertices that includes at least one endpoint of every edge of the graph. The problem of finding a minimum vertex cover is a classical optimization problem in computer science and is a typical example of an NP-hard optimization problem that has an approximation algorithm. Its decision version, the vertex cover problem, was one of Karp's 21 NP-complete problems and is therefore a classical NP-complete problem in computational complexity theory. Furthermore, the vertex cover problem is fixed-parameter tractable and a central problem in parameterized complexity theory.
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In graph theory, a cograph, or complement-reducible graph, or P4-free graph, is a graph that can be generated from the single-vertex graph K1 by complementation and disjoint union. That is, the family of cographs is the smallest class of graphs that includes K1 and is closed under complementation and disjoint union.
In graph theory, a dominating set for a graph G = (V, E) is a subset D of V such that every vertex not in D is adjacent to at least one member of D. The domination number γ(G) is the number of vertices in a smallest dominating set for G.
In graph theory, a domatic partition of a graph is a partition of into disjoint sets , ,..., such that each Vi is a dominating set for G. The figure on the right shows a domatic partition of a graph; here the dominating set consists of the yellow vertices, consists of the green vertices, and consists of the blue vertices.
In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set. In other words, there is no vertex outside the independent set that may join it because it is maximal with respect to the independent set property.
In the mathematical discipline of graph theory, a feedback vertex set of a graph is a set of vertices whose removal leaves a graph without cycles. In other words, each feedback vertex set contains at least one vertex of any cycle in the graph. The feedback vertex set problem is an NP-complete problem in computational complexity theory. It was among the first problems shown to be NP-complete. It has wide applications in operating systems, database systems, and VLSI chip design.
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In graph theory and theoretical computer science, a maximum common induced subgraph of two graphs G and H is a graph that is an induced subgraph of both G and H, and that has as many vertices as possible.
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In graph theory, a clique cover or partition into cliques of a given undirected graph is a partition of the vertices of the graph into cliques, subsets of vertices within which every two vertices are adjacent. A minimum clique cover is a clique cover that uses as few cliques as possible. The minimum k for which a clique cover exists is called the clique cover number of the given graph.
In graph theory and theoretical computer science, the longest path problem is the problem of finding a simple path of maximum length in a given graph. A path is called simple if it does not have any repeated vertices; the length of a path may either be measured by its number of edges, or by the sum of the weights of its edges. In contrast to the shortest path problem, which can be solved in polynomial time in graphs without negative-weight cycles, the longest path problem is NP-hard and the decision version of the problem, which asks whether a path exists of at least some given length, is NP-complete. This means that the decision problem cannot be solved in polynomial time for arbitrary graphs unless P = NP. Stronger hardness results are also known showing that it is difficult to approximate. However, it has a linear time solution for directed acyclic graphs, which has important applications in finding the critical path in scheduling problems.
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