Maximal independent set

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In graph theory, a maximal independent set or maximal stable set is an independent set that is not a subset of any other independent set. That is, it is a set S such that every edge of the graph has at least one endpoint not in S and every vertex not in S has at least one neighbor in S. A maximal independent set is also a dominating set in the graph, and every dominating set that is independent must be maximal independent, so maximal independent sets are also called independent dominating sets. A graph may have many maximal independent sets, of widely varying sizes^[1]; the largest maximal independent set is called the maximum independent set.

For example, in the graph P₃, a path with three vertices a, b, and c, and two edges ab and bc, the sets {b} and {a,c} are both maximally independent. The set {a} is independent, but is not maximal independent, because it is a subset of the larger independent set {a,c}. In this same graph, the maximal cliques are the sets {a,b} and {b,c}.

The phrase "maximal independent set" is also used to describe maximal subsets of independent elements in mathematical structures other than graphs, and in particular in matroids.

1 Related vertex sets
2 Graph family characterizations
3 Bounding the number of sets
4 Set listing algorithms
5 Notes
6 References

[edit] Related vertex sets

If S is a maximal independent set in some graph, it is a maximal clique or maximal complete subgraph in the complement graph. A maximal clique is a set of vertices that induces a complete subgraph, and that is not a subset of the vertices of any larger complete subgraph. That is, it is a set S such that every pair of vertices in S is connected by an edge and every vertex not in S is missing an edge to at least one vertex in S. A graph may have many maximal cliques, of varying sizes; the largest of these is the maximum clique. Some authors include maximality as part of the definition of a clique, and refer to maximal cliques simply as cliques.

The complement of a maximal independent set, that is, the vertices not belonging to the set, forms a minimal vertex cover. That is, the complement is a vertex cover, a set of vertices that includes at least one endpoint of each edge, such that no other vertex cover is a subset of it. Minimal vertex covers have been studied in statistical mechanics in connection with the hard-sphere lattice gas model, a mathematical abstraction of fluid-solid state transitions.^[2]

[edit] Graph family characterizations

Certain graph families have also been characterized in terms of their maximal cliques or maximal independent sets. Examples include the maximal clique irreducible and hereditary maximal clique irreducible graphs. A graph is said to be maximal clique irreducible if every maximal clique has an edge that belongs to no other maximal clique, and hereditary maximal clique irreducible if the same property is true for every induced subgraph.^[3] Hereditary maximal clique irreducible graphs include triangle-free graphs and bipartite graphs, and interval graphs.

Cographs can be characterized as graphs in which every maximal clique intersects every maximal independent set, and in which the same property is true in all induced subgraphs.

[edit] Bounding the number of sets

Moon and Moser (1965) showed that any graph with n vertices has at most 3^n/3 maximal cliques. Complementarily, any such graph also has at most 3^n/3 maximal independent sets. A graph with exactly 3^n/3 maximal independent sets is easy to construct: simply take the disjoint union of n/3 triangle graphs. Any maximal independent set in this graph is formed by choosing one vertex from each triangle. The complementary graph, with exactly 3^n/3 maximal cliques, is a special type of Turán graph; because of their connection with Moon and Moser's bound, these graphs are also sometimes called Moon-Moser graphs. Tighter bounds are possible if one limits the size of the maximal independent sets: the number of maximal independent sets of size k in any n-vertex graph is at most

$\lfloor n/k \rfloor^{k-(n\bmod k)}\lfloor n/k+1 \rfloor^{n\bmod k}.$

The graphs achieving this bound are again Turán graphs.^[4]

Certain families of graphs may, however, have much more restrictive bounds on the numbers of maximal independent sets or maximal cliques. For instance, if all graphs in a family of graphs have O(n) edges, and the family is closed under subgraphs, then all maximal cliques have constant size and there can be at most linearly many maximal cliques.^[5]

Any maximal clique irreducible graph, clearly, has at most as many maximal cliques as it has edges. A tighter bound is possible for interval graphs, and more generally chordal graphs: in these graphs there can be at most n maximal cliques.

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.^[6] Therefore, both numbers are proportional to powers of 1.324718, the plastic number.

[edit] Set listing algorithms

An algorithm for listing all maximal independent sets or maximal cliques in a graph can be used as a subroutine for solving many NP-complete graph problems. Most obviously, the solutions to the maximum independent set problem, the maximum clique problem, and the minimum independent dominating problem must all be maximal independent sets or maximal cliques, and can be found by an algorithm that lists all maximal independent sets or maximal cliques and retains the ones with the largest or smallest size. Similarly, the minimum vertex cover can be found as the complement of one of the maximal independent sets. Lawler (1976) observed that listing maximal independent sets can also be used to find 3-colorings of graphs: a graph can be 3-colored if and only if the complement of one of its maximal independent sets is bipartite. He used this approach not only for 3-coloring but as part of a more general graph coloring algorithm, and similar approaches to graph coloring have been refined by other authors since.^[7] Other more complex problems can also be modeled as finding a clique or independent set of a specific type. This motivates the algorithmic problem of listing all maximal independent sets (or equivalently, all maximal cliques) efficiently.

It is straightforward to turn a proof of Moon and Moser's 3^n/3 bound on the number of maximal independent sets into an algorithm that lists all such sets in time O(3^n/3).^[8] For graphs that have the largest possible number of maximal independent sets, this algorithm takes constant time per output set. However, an algorithm with this time bound can be highly inefficient for graphs with more limited numbers of independent sets. For this reason, many researchers have studied algorithms that list all maximal independent sets in polynomial time per output set.^[9] The time per maximal independent set is proportional to that for matrix multiplication in dense graphs, or faster in various classes of sparse graphs.^[10]

[edit] Notes

^ Erdős (1966) shows that the number of different sizes of maximal independent sets in an n-vertex graph may be as large as n - log n - O(log log n) and is never larger than n - log n.
^ Weigt and Hartmann (2001).
^ Information System on Graph Class Inclusions: maximal clique irreducible graphs and hereditary maximal clique irreducible graphs.
^ Byskov (2003). For related earlier results see Croitoru (1979) and Eppstein (2003).
^ Chiba and Nishizeki (1985). The sparseness condition is equivalent to assuming that the graph family has bounded arboricity.
^ Bisdorff and Marichal (2007); Euler (2005); Füredi (1987).
^ Eppstein (2003), Byskov (2003).
^ Eppstein (2003).
^ Bomze et al. (1999); Eppstein (2005); Jennings and Motycková (1992); Johnson et al. (1988); Lawler et al. (1980); Liang et al. (1991); Makino and Uno (2004); Mishra and Pitt (1997); Stix (2004); Tsukiyama et al. (1977); Yu and Chen (1993).
^ Makino and Uno (2004); Eppstein (2005).

[edit] References

Bisdorff, R.; Marichal, J.-L. (2007). "Counting non-isomorphic maximal independent sets of the n-cycle graph". arXiv:math.CO/0701647.

Bomze, I. M.; Budinich, M.; Pardalos, P. M.; Pelillo, M. (1999). "The maximum clique problem". Handbook of Combinatorial Optimization, vol. 4: 1–74, Kluwer.

Byskov, J. M. (2003). "Algorithms for k-colouring and finding maximal independent sets". Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms: 456–457.

Chiba, N.; Nishizeki, T. (1985). "Arboricity and subgraph listing algorithms". SIAM J. on Computing 14 (1): 210–223. DOI:10.1137/0214017.

Croitoru, C. (1979). "On stables in graphs". Proc. Third Coll. Operations Research: 55–60, Babeş-Bolyai University, Cluj-Napoca, Romania.

Eppstein, D. (2003). "Small maximal independent sets and faster exact graph coloring". Journal of Graph Algorithms and Applications 7 (2): 131–140. arXiv:cs.DS/cs.DS/0011009.

Eppstein, D. (2005). "All maximal independent sets and dynamic dominance for sparse graphs". Proc. Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms: 451–459.

Erdős, P. (1966). "On cliques in graphs". Israel J. Math. 4: 233–234. MR 0205874.

Euler, R. (2005). "The Fibonacci number of a grid graph and a new class of integer sequences". Journal of Integer Sequences 8 (2): 05.2.6.

Füredi, Z. (1987). "The number of maximal independent sets in connected graphs". Journal of Graph Theory 11 (4): 463–470.

Jennings, E.; Motycková, L. (1992). "A distributed algorithm for finding all maximal cliques in a network graph". Proc. First Latin American Symposium on Theoretical Informatics: 281–293, Lecture Notes in Computer Science, vol. 583, Springer-Verlag.

Johnson, D. S.; Yannakakis, M.; Papadimitriou, C. H. (1988). "On generating all maximal independent sets". Information Processing Letters 27 (3): 119–123.

Lawler, E. L. (1976). "A note on the complexity of the chromatic number problem". Information Processing Letters 5 (3): 66–67.

Lawler, E. L.; Lenstra, J. K.; Rinnooy Kan, A. H. G. (1980). "Generating all maximal independent sets: NP-hardness and polynomial time algorithms". SIAM J. Computing 9 (3): 558–565.

Leung, J. Y.-T. (1984). "Fast algorithms for generating all maximal independent sets of interval, circular-arc and chordal graphs". Journal of Algorithms 5: 22–35.

Liang, Y. D.; Dhall, S. K.; Lakshmivarahan, S. (1991). "On the problem of finding all maximum weight independent sets in interval and circular arc graphs". Proc. Symp. Applied Computing: 465–470.

Makino, K.; Uno, T. (2004). "New algorithms for enumerating all maximal cliques". Proc. Ninth Scandinavian Workshop on Algorithm Theory: 260–272.

Mishra, N.; Pitt, L. (1997). "Generating all maximal independent sets of bounded-degree hypergraphs". Proc. Tenth Conf. Computational Learning Theory: 211–217.

Moon, J. W.; Moser, L. (1965). "On cliques in graphs". Israel J. Math. 3: 23–28. MR 0182577.

Stix, V. (2004). "Finding all maximal cliques in dynamic graphs". Computational Optimization Appl. 27 (2): 173–186.

Tsukiyama, S.; Ide, M.; Ariyoshi, H.; Shirakawa, I. (1977). "A new algorithm for generating all the maximal independent sets". SIAM J. on Computing 6 (3): 505–517.

Weigt, Martin; Hartmann, Alexander K. (2001). "Minimal vertex covers on finite-connectivity random graphs: A hard-sphere lattice-gas picture". Phys. Rev. E 63 (5): 056127. DOI:10.1103/PhysRevE.63.056127. arXiv:cond-mat/0011446.