Induced Matching
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Induced Matching

In graph theory, an induced matching or strong matching is a subset of the edges of an undirected graph that do not share any vertices (it is a matching) and includes every edge connecting any two vertices in the subset (it is an induced subgraph).

An induced matching can also be described as an independent set in the square of the line graph of the given graph.[1]

Strong coloring and neighborhoods

The minimum number of induced matchings into which the edges of a graph can be partitioned is called its strong chromatic index, by analogy with the chromatic index of the graph, the minimum number of matchings into which its edges can be partitioned.[2] It equals the chromatic number of the square of the line graph. Brooks' theorem, applied to the square of the line graph, shows that the strong chromatic index is at most quadratic in the maximum degree of the given graph, but better constant factors in the quadratic bound can be obtained by other methods.[3]

The Ruzsa-Szemerédi problem concerns the edge density of balanced bipartite graphs with linear strong chromatic index. Equivalently, it concerns the density of a different class of graphs, the locally linear graphs in which the neighborhood of every vertex is an induced matching.[4] Neither of these types of graph can have a quadratic number of edges, but constructions are known for graphs of this type with nearly-quadratic numbers of edges.[5]

Computational complexity

Finding an induced matching of size at least ${\displaystyle k}$ is NP-complete (and thus, finding an induced matching of maximum size is NP-hard). It can be solved in polynomial time in chordal graphs, because the squares of line graphs of chordal graphs are perfect graphs.[6] Moreover, it can be solved in linear time in chordal graphs [7]. Unless an unexpected collapse in the polynomial hierarchy occurs, the largest induced matching cannot be approximated to within any ${\displaystyle n^{1-\varepsilon }}$ approximation ratio in polynomial time.[8]

The problem is also W[1]-hard, meaning that even finding a small induced matching of a given size ${\displaystyle k}$ is unlikely to have an algorithm significantly faster than the brute force search approach of trying all ${\displaystyle k}$-tuples of edges.[9] However, the problem of finding ${\displaystyle k}$ vertices whose removal leaves an induced matching is fixed-parameter tractable.[10] The problem can also be solved exactly on ${\displaystyle n}$-vertex graphs in time ${\displaystyle O(1.3752^{n})}$ with exponential space, or in time ${\displaystyle O(1.4231^{n})}$ with polynomial space.[11]

References

1. ^ Cameron, Kathie (2004), "Induced matchings in intersection graphs", Discrete Mathematics, 278 (1-3): 1-9, doi:10.1016/j.disc.2003.05.001, MR 2035386
2. ^ Fouquet, J.-L.; Jolivet, J.-L. (1983), "Strong edge-colorings of graphs and applications to multi-k-gons", Ars Combinatoria, 16 (A): 141-150, MR 0737086
3. ^ Molloy, Michael; Reed, Bruce (1997), "A bound on the strong chromatic index of a graph", Journal of Combinatorial Theory, Series B, 69 (2): 103-109, doi:10.1006/jctb.1997.1724, hdl:1807/9474, MR 1438613
4. ^ Fron?ek, Dalibor (1989), "Locally linear graphs", Mathematica Slovaca, 39 (1): 3-6, hdl:10338.dmlcz/136481, MR 1016323
5. ^ Ruzsa, I. Z.; Szemerédi, E. (1978), "Triple systems with no six points carrying three triangles", Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, Colloq. Math. Soc. János Bolyai, 18, Amsterdam and New York: North-Holland, pp. 939-945, MR 0519318
6. ^ Cameron, Kathie (2008), "Maximum Induced Matchings for Chordal Graphs in Linear Time", Special issue for First Montreal Conference on Combinatorics and Computer Science, 1987, Algorithmica, 52: 440-447, doi:10.1016/0166-218X(92)90275-F, MR 1011265
7. ^ Brandstaedt, Andreas; Hoang, Chinh (1989), "Induced matchings", Discrete Applied Mathematics, 24 (1-3): 97-102, doi:10.1007/s00453-007-9045-2
8. ^ Chalermsook, Parinya; Laekhanukit, Bundit; Nanongkai, Danupon (2012), "Graph products revisited: tight approximation hardness of induced matching, poset dimension and more", Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, Philadelphia, Pennsylvania: SIAM, pp. 1557-1576, MR 3202998
9. ^ Moser, Hannes; Sikdar, Somnath (2009), "The parameterized complexity of the induced matching problem", Discrete Applied Mathematics, 157 (4): 715-727, doi:10.1016/j.dam.2008.07.011, MR 2499485
10. ^ Xiao, Mingyu; Kou, Shaowei (2016), "Almost induced matching: linear kernels and parameterized algorithms", in Heggernes, Pinar (ed.), Graph-Theoretic Concepts in Computer Science: 42nd International Workshop, WG 2016, Istanbul, Turkey, June 22-24, 2016, Revised Selected Papers, Lecture Notes in Computer Science, 9941, Berlin: Springer, pp. 220-232, doi:10.1007/978-3-662-53536-3_19, MR 3593958
11. ^ Xiao, Mingyu; Tan, Huan (2017), "Exact algorithms for maximum induced matching", Information and Computation, 256: 196-211, doi:10.1016/j.ic.2017.07.006, MR 3705425