Abstract
A graph G = (V,E) is a tolerance graph if each vertex v ∈ V can be associated with an interval of the real line I v and a positive real number t v in such a way that (uv) ∈ E if and only if |I v ∩ I u | ≥ min (t v ,t u ). No algorithm for recognizing tolerance graphs in general is known. In this paper we present an O(n + m) algorithm for recognizing tolerance graphs that are also bipartite, where n and m are the number vertices and edges of the graph, respectively. We also give a new structural characterization of these graphs based on the algorithm.
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References
Booth, K., Lueker, G.: Linear Algorithms to Recognize Interval Graphs and Test for the Consecutive Ones Property. In: Proceedings of the Seventh Annual ACM Symposium on Theory of Computing, pp. 255–265 (1975)
Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes, A Survey. Soc. for Industrial and Applied Math., Philadelphia, PA (1999)
Brandstädt, A., Spinrad, J.P., Stewart, L.: Bipartite permutation graphs are bipartite tolerance graphs. Congress. Numer. 58, 165–174 (1987)
Brown, D.E.: Variations on Interval Graphs, Ph.D. thesis, University of Colorado at Denver (2004)
Busch, A.H.: A characterization of bipartite tolerance graphs. Discrete Applied Math. 154, 471–477 (2006)
Chang, G., Kloks, A., Liu, J., Peng, S.: The PIGS Full Monty - A Floor Show of Minimal Separators. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 521–532. Springer, Heidelberg (2005)
Derigs, U., Goecke, O., Schrader, R.: Bisimplicial edges, gaussian elimination and matchings in bipartite graphs. In: Inter. Workshop on Graph-Theoretic Concepts in Comp. Sci., pp. 79–87 (1984)
Golumbic, M.C., Monma, C.L.: A generalization of interval graphs with tolerances. Congress. Numer. 35, 321–331 (1982)
Golumbic, M.C., Monma, C.L., Trotter, W.T.: Tolerance Graphs. Discrete Applied Math. 9, 157–170 (1984)
Golumbic, M.C., Trenk, A.N.: Tolerance Graphs. Cambridge University Press, Cambridge, UK (2004)
Hayward, R.B., Shamir, R.: A note on tolerance graph recognition. Discrete Appl. Math. 143, 307–311 (2004)
Langley, L.: Interval tolerance orders and dimension, Ph.D. thesis, Dartmouth College (1993)
Lubiw, A.: Doubly lexical orderings of matrices. Siam J. Comput. 16, 854–879 (1987)
Müller, H.: Recognizing Interval digraphs and interval bigraphs in polynomial time. Discrete Appl. Math. 78, 189–205 (1997)
Paige, R., Tarjan, R.E.: Three partition refinements of algorithms. SIAM J. Comput. 16, 973–989 (1987)
Sheng, L.: Cycle free probe interval graphs. Congr. Numer. 140, 33–42 (1999)
Spinrad, J.P.: Doubly lexical ordering of dense 0-1 matrices. Inf. Proc. Lett. 45, 229–235 (1993)
Spinrad, J.P., Brandstädt, A., Stewart, L.: Bipartite permutation graphs. Discrete Appl. Math. 18, 279–292 (1987)
Tarjan, R.E.: Depth-First Search and Linear Graph Algorithms. SIAM J. Comput. 1, 146–160 (1972)
Tucker, A.C.: A Structure Theorem for the Consecutive 1’s Property. J. Combinatorial Theory Ser. B 12, 153–162 (1972)
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Busch, A.H., Isaak, G. (2007). Recognizing Bipartite Tolerance Graphs in Linear Time. In: Brandstädt, A., Kratsch, D., Müller, H. (eds) Graph-Theoretic Concepts in Computer Science. WG 2007. Lecture Notes in Computer Science, vol 4769. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74839-7_2
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DOI: https://doi.org/10.1007/978-3-540-74839-7_2
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