Abstract
The Maximum Induced Matching (MIM) Problem asks for a largest set of pairwise vertex-disjoint edges in a graph which are pairwise of distance at least two. It is well-known that the MIM problem is NP-complete even on particular bipartite graphs and on line graphs. On the other hand, it is solvable in polynomial time for various classes of graphs (such as chordal, weakly chordal, interval, circular-arc graphs and others) since the MIM problem on graph G corresponds to the Maximum Independent Set problem on the square G *=L(G)2 of the line graph L(G) of G, and in some cases, G * is in the same graph class; for example, for chordal graphs G, G * is chordal. The construction of G *, however, requires \({\mathcal{O}}(m^{2})\) time, where m is the number of edges in G. Is has been an open problem whether there is a linear-time algorithm for the MIM problem on chordal graphs. We give such an algorithm which is based on perfect elimination order and LexBFS.
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Brandstädt, A., Dragan, F.F., Nicolai, F.: LexBFS-orderings and powers of chordal graphs. Discrete Math. 171, 27–42 (1997)
Cameron, K.: Induced matchings. Discrete Appl. Math. 24, 97–102 (1989)
Cameron, K.: Induced matchings in intersection graphs (extended abstract). Electron. Notes Discrete Math. 5 (2000)
Cameron, K., Sritharan, R., Tang, Y.: Finding a maximum induced matching in weakly chordal graphs. Discrete Math. 266, 133–142 (2003)
Chang, J.-M.: Induced matchings in asteroidal-triple-free graphs. Discrete Appl. Math. 132, 67–78 (2004)
Fricke, G., Laskar, R.: Strong matchings on trees. Congr. Numer. 89, 239–243 (1992)
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic, San Diego (1980)
Golumbic, M.C., Laskar, R.C.: Irredundancy in circular-arc graphs. Discrete Appl. Math. 44, 79–89 (1993)
Golumbic, M.C., Lewenstein, M.: New results on induced matchings. Discrete Appl. Math. 101, 157–165 (2000)
Ko, C.W., Shepherd, F.B.: Bipartite domination and simultaneous matroid covers. SIAM J. Discrete Math. 16, 517–523 (2003)
Kobler, D., Rotics, U.: Finding maximum induced matchings in subclasses of claw-free and P 5-free graphs, and in graphs with matching and induced matching of equal maximum size. Algorithmica 37, 327–346 (2003)
Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5, 266–283 (1976)
Stockmeyer, L.J., Vazirani, V.V.: NP-completeness of some generalizations of the maximum matching problem. Inform. Process. Lett. 15, 14–19 (1982)
Zito, M.: Linear time maximum induced matching algorithm for trees. Nord. J. Comput. 7, 58–63 (2000)
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Brandstädt, A., Hoàng, C.T. Maximum Induced Matchings for Chordal Graphs in Linear Time. Algorithmica 52, 440–447 (2008). https://doi.org/10.1007/s00453-007-9045-2
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DOI: https://doi.org/10.1007/s00453-007-9045-2