# Using bounded degree spanning trees in the design of efficient algorithms on claw-free graphs

## Abstract

Claw-free graphs are graphs that do not have *K*_{1,3} as an induced subgraph. Line graphs, a special case of claw-free graphs, are the intersection graphs of edges in simple graphs. We show how to compute efficiently in parallel a binary tree that will be a rooted spanning tree of the claw-free graph. Every binary tree contains at least one edge whose removal partitions the tree into two subtrees of nearly equal cardinality, and this separator can be found efficiently in parallel. We solve problems on claw-free graphs by a divide-and-conquer strategy. The advantage of our partition is that the vertices in each set induce a connected subgraph. The problems are solved for the two subgraphs, and then the results are combined to get a solution for the entire graph.

Both the problem of finding a perfect matching in claw-free graphs and the problem of reconstructing a root graph from a line graph are amenable to this approach. We present a nearly optimal parallel NC algorithm for computing a perfect matching that runs in time *O*(log^{2}*n*) with *O*(*n*+*m*) processors on an EREW PRAM. Also, we present a sequential linear time algorithm to compute this perfect matching. Then, we present an efficient parallel reconstruction of root graphs from line graphs. If *G*=(*V, E*) denotes a line graph, then the algorithm runs in *O*(log ¦*V*¦) time using *O*(¦*E*¦) processors in the CRCW PRAM model. It is optimal up to a polylogarithmic factor. Previously, it was known how to reconstruct the root graph in *NC* using a large (though polynomial) number of processors; this is the first algorithm that employs a linear number of processors.

## Preview

Unable to display preview. Download preview PDF.

## References

- [BT]M.Ben-Or and P.Tiwari, A deterministic algorithm for sparse multivariate polynomial interpolation,
*Proc. 20th Symposium on Theory on Computing*, 1988, pp. 301–309.Google Scholar - [Ed]J. Edmonds, Paths, trees and flowers,
*Canad. J. Math.*17 (1965) 449–467.Google Scholar - [GK]D. Y. Grigoriev and M. Karpiński, The matching problem for bipartite graphs with polynomially bounded permanents is in
*NC, 28th Annual IEEE Symposium on Foundations of Computer Science*, Los Angeles, CA, 1987, pp. 166–172.Google Scholar - [Ha]F. Harary,
*Graph Theory*, Addison-Wesley Publishing Company, 1972.Google Scholar - [He]Xin He, Private communication.Google Scholar
- [Jo1]D.S. Johnson, The NP-completeness column: an ongoing guide,
*J. Algorithms*6 (1985) 434–451.CrossRefGoogle Scholar - [Jo2]D.S. Johnson, The NP-completeness column: an ongoing guide,
*J. Algorithms*8 (1987) 438–448.CrossRefGoogle Scholar - [KUW]R. M. Karp, E. Upfal and A. Wigderson, Constructing a perfect matching is in random
*NC*,*Combinatorica*6, 1 (1986) 35–48.Google Scholar - [La]M. Las Vergnas, A note on matchings in graphs,
*Colloque sur la Théorie des Graphes (Paris 1974)*, Cahiers Centre Etudes Rech. Opér. 17, 1975, 257–260.Google Scholar - [Le]P. G. H. Lehot, An optimal algorithm to detect a line graph and output its root graph,
*JACM***21**, (1974) pp. 569–575.Google Scholar - [Mi]G.J. Minty, On maximal independent sets of vertices in claw-free graphs,
*J. Comb. Theory B*28 (1980) 284–304.Google Scholar - [MN]G.Miller and J.Naor, Flow in planar graphs with multiple sources and sinks,
*manuscript*.Google Scholar - [MR]G. L. Miller and J. Reif, Parallel tree contraction and its applications,
*Proceedings 26th Annual IEEE Symposium on Foundations of Computer Science*, (1985) pp. 478–489.Google Scholar - [MVV]K. Mulmuley, U. V. Vazirani and V. V. Vazirani, Matching is as easy as matrix inversion,
*Combinatorica*7, 1 (1987) 105–113.Google Scholar - [Na]J. Naor, Computing a perfect matching in a line graph,
*3rd International Workshop on Parallel Computation and VLSI Theory, Aegean Workshop on Computing*, Corfu Island, Greece (1988).Google Scholar - [Ro]N. D. Roussopoulos, A max {
*m, n*} algorithm for determining the graph H from its line graph G,*Information Processing Letters***2**(1973) pp. 108–112.Google Scholar - [Sb]N. Sbihi, Algorithme de recherche d'un stable de cardinalité maximum dans un graphe,
*Disc. Math.*29 (1980) 53–76.Google Scholar - [Su]D. P. Summer, On Tutte's factorization theorem,
*Proceedings of the Capital Conference on Graph Theory and Combinatorics*, George Washington University, Washington D.C., 1973, pp. 350–355. Lecture Notes in Math., Vol. 406, Springer, Berlin, 1974.Google Scholar - [SV]Y. Shiloach and U. Vishkin, An
*O*(log*n*) parallel connectivity algorithm,*J. Algorithms***3**(1982) 57–67.CrossRefGoogle Scholar - [Wh]H. Whitney, Congruent graphs and the connectivity of graphs,
*Amer. J. Math.***54**(1932), 150–168.Google Scholar