Abstract
A graph is HHD-free is it does not contain a house (i.e., the complement of P 5), a hole (a cycle of length at least 5) or a domino (the graph obtained from two 4-cycles by identifying an edge in one C 4 with an edge in the other C 4) as an induced subgraph. The minimum fill-in problem is the problem of finding a chordal supergraph with the smallest possible number of edges. The treewidth problem is the problem of finding a chordal embedding of the graph with the smallest possible clique number. In this note we show that both problems are solvable in polynomial time for HHD-free graphs.
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Arnborg, S., D. G. Corneil and A. Proskurowski, Complexity of finding embeddings in a k-tree, SIAM J. Alg. Disc. Meth. 8, (1987), pp. 277–284.
Bodlaender, H., T. Kloks, D. Kratsch and H. Müller, Treewidth and minimum fill-in on d-trapezoid graphs, Technical report RUU-CS-1995-34, Utrecht University, The Netherlands, 1995.
Bodlaender, H. and R. Möhring, The pathwidth and treewidth of cographs, SIAM Journal on Discrete Mathematics 7 (1993), pp. 181–188.
Brandstädt, A., Special graph classes — A survey, Schriftenreihe des Fachbereichs Mathematik, SM-DU-199 (1991), Universität Duisburg Gesamthochschule.
Maw-Shang Chang, Algorithms for maximum matching and minimum fill-in on chordal bipartite graphs. ISAAC'96 (T. Asano et al. ed.), LLNCS 1178, pp. 146–155.
Dirac, G. A., On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25, (1961), pp. 71–76.
Golumbic M. C., Algorithmic graph theory and perfect graphs, Academic Press, New York, 1980.
Hammer, P. L. and F. Maffray, Completely separable graphs, Discrete Applied Mathematics 27, (1990), pp. 85–99.
Hayward, R., C. T. Hoang and F. Maffray, Optimizing weakly triangulated graphs, Graphs and combinatorics 5, (1989), pp. 339–349.
Jamison, B. and S. Olariu, On the semi-perfect elimination, Advances in Applied Mathematics 9, (1988), pp. 364–376.
Kloks, T., Treewidth — Computations and Approximations, Springer Verlag, Lecture Notes in Computer Science 842, (1994).
Kloks, T., Treewidth of circle graphs, International Journal of Foundations of Computer Science 7, (1996), pp. 111–120.
Kloks, T. and D. Kratsch, Treewidth of chordal bipartite graphs, J. of Algorithms 19, (1995), pp. 266–281.
Kloks, T., D. Kratsch and H. Müller, Approximating the bandwidth for AT-free graphs, Proceedings of the Third Annual European Symposium on Algorithms (ESA'95), Springer-Verlag, Lecture Notes in Computer Science 979, (1995), pp. 434–447.
Kloks, T., D. Kratsch and J. Spinrad, Treewidth and pathwidth of cocomparability graphs of bounded dimension, Computing Science Notes, 93/46, Eindhoven University of Technology, Eindhoven, The Netherlands, (1993), to appear in Order.
Kloks, T., D. Kratsch and C. K. Wong, Minimum fill-in of circle and circular arc graphs, Proceedings of the 21 th International Symposium on Automata, Languages and Programming (ICALP'96), Springer-Verlag Lecture Notes in Computer Science 1113, (1996), pp. 256–267.
Olariu, S., Results on perfect graphs, PhD thesis, Scool of Computer Science, McGill University, Montreal, 1986.
Parra, A., Scheffler, P., How to use minimal separators for its chordal triangulation, ICALP'95, LLNCS 944, pp. 123–134.
Spinrad, J., A. Brandstädt and L. Stewart, Bipartite permutation graphs, Discrete Applied mathematics 18, (1987), pp. 279–292.
Sundaram, R., K. Sher Singh and C. Pandu Rangan, Treewidth of circular arc graphs. To appear in SIAM J. Disc. Math.
Yannakakis, M., Computing the minimum fill-in is NP-complete, SIAM J. Alg. Disc. Meth. 2, (1981), pp. 77–79.
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Broersma, H.J., Dahlhaus, E., Kloks, T. (1997). Algorithms for the treewidth and minimum fill-in of HHD-free graphs. In: Möhring, R.H. (eds) Graph-Theoretic Concepts in Computer Science. WG 1997. Lecture Notes in Computer Science, vol 1335. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0024492
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