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# Efficiently Realizing Interval Sequences

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LIPIcs.ISAAC.2019.47.pdf
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## Cite As

Amotz Bar-Noy, Keerti Choudhary, David Peleg, and Dror Rawitz. Efficiently Realizing Interval Sequences. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 47:1-47:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ISAAC.2019.47

## Abstract

We consider the problem of realizable interval-sequences. An interval sequence comprises of n integer intervals [a_i,b_i] such that 0 <= a_i <= b_i <= n-1, and is said to be graphic/realizable if there exists a graph with degree sequence, say, D=(d_1,...,d_n) satisfying the condition a_i <= d_i <= b_i, for each i in [1,n]. There is a characterisation (also implying an O(n) verifying algorithm) known for realizability of interval-sequences, which is a generalization of the Erdös-Gallai characterisation for graphic sequences. However, given any realizable interval-sequence, there is no known algorithm for computing a corresponding graphic certificate in o(n^2) time. In this paper, we provide an O(n log n) time algorithm for computing a graphic sequence for any realizable interval sequence. In addition, when the interval sequence is non-realizable, we show how to find a graphic sequence having minimum deviation with respect to the given interval sequence, in the same time. Finally, we consider variants of the problem such as computing the most regular graphic sequence, and computing a minimum extension of a length p non-graphic sequence to a graphic one.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Graph algorithms
• Mathematics of computing → Enumeration
##### Keywords
• Graph realization
• graphic sequence
• interval sequence

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## References

1. Martin Aigner and Eberhard Triesch. Realizability and uniqueness in graphs. Discrete Mathematics, 136:3-20, 1994.
2. Richard Anstee. Properties of a class of (0,1)-matrices covering a given matrix. Can. J. Math., pages 438-453, 1982.
3. Richard Anstee. An algorithmic proof of Tutte’s f-factor theorem. J. Algorithms, 6(1):112-131, 1985.
4. M. Behzad and James E. Simpson. Eccentric sequences and eccentric sets in graphs. Discrete Mathematics, 16(3):187-193, 1976.
5. Mao-cheng Cai, Xiaotie Deng, and Wenan Zang. Solution to a problem on degree sequences of graphs. Discrete Mathematics, 219(1-3):253-257, 2000.
6. Wai-Kai Chen. On the realization of a (p,s)-digraph with prescribed degrees. Journal of the Franklin Institute, 281(5):406-422, 1966.
7. Paul Erdös and Tibor Gallai. Graphs with Prescribed Degrees of Vertices [Hungarian]. Matematikai Lapok, 11:264-274, 1960.
8. A. Frank. Augmenting graphs to meet edge-connectivity requirements. SIAM J. Discrete Math., 5:25-43, 1992.
9. A. Frank. Connectivity augmentation problems in network design. In Mathematical Programming: State of the Art, pages 34-63. Univ. Michigan, 1994.
10. H. Frank and W. Chou. Connectivity considerations in the design of survivable networks. IEEE Trans. Circuit Theory, CT-17:486-490, 1970.
11. Satoru Fujishige and Sachin B. Patkar. Realization of set functions as cut functions of graphs and hypergraphs. Discrete Mathematics, 226(1-3):199-210, 2001.
12. D.R. Fulkerson. Zero-one matrices with zero trace. Pacific J. Math., 12:831-836, 1960.
13. Ankit Garg, Arpit Goel, and Amitabha Tripathi. Constructive extensions of two results on graphic sequences. Discrete Applied Mathematics, 159(17):2170-2174, 2011.
14. R.E. Gomory and T.C. Hu. Multi-terminal network flows. J. Soc. Industrial & Applied Math., 9, 1961.
15. Jiyun Guo and Jianhua Yin. A variant of Niessen’s problem on degree sequences of graphs. Discrete Mathematics and Theoretical Computer Science, Vol. 16 no. 1 (in progress)(1):287-292, May 2014. Graph Theory. URL: https://hal.inria.fr/hal-01179211.
16. S. Louis Hakimi. On Realizability of a Set of Integers as Degrees of the Vertices of a Linear Graph -I. SIAM J. Appl. Math., 10(3):496-506, 1962.
17. Sepp Hartung and André Nichterlein. NP-Hardness and Fixed-Parameter Tractability of Realizing Degree Sequences with Directed Acyclic Graphs. SIAM J. Discrete Math., 29(4):1931-1960, 2015.
18. V. Havel. A Remark on the Existence of Finite Graphs [in Czech]. Casopis Pest. Mat., 80:477-480, 1955.
19. Katherine Heinrich, Pavol Hell, David G. Kirkpatrick, and Guizhen Liu. A simple existence criterion for g < f–factors, with applications to [a, b]–factors. Discrete Mathematics, 85:313-317, 1990.
20. Pavol Hell and David G. Kirkpatrick. Linear-time certifying algorithms for near-graphical sequences. Discrete Mathematics, 309(18):5703-5713, 2009.
21. Daniel J. Kleitman and D. L. Wang. Algorithms for constructing graphs and digraphs with given valences and factors. Discrete Mathematics, 6(1):79-88, 1973.
22. Linda Lesniak. Eccentric sequences in graphs. Periodica Mathematica Hungarica, 6(4):287-293, 1975.
23. László Lovász. Subgraphs with prescribed valencies. J. Comb. Theory, 8:391-416, 1970.
24. N.V.R. Mahadev and U.N. Peled. Threshold Graphs and Related Topics. Annals of Discrete Mathematics. Elsevier Science, 1995.
25. Akutsu Tatsuya and Hiroshi Nagamochi. Comparison and enumeration of chemical graphs. Computational and structural biotechnology, 5, 2013.
26. Amitabha Tripathi, Sushmita Venugopalan, and Douglas B. West. A short constructive proof of the Erdos-Gallai characterization of graphic lists. Discrete Mathematics, 310(4):843-844, 2010.
27. W. T. Tutte. Graph factors. Combinatorica, 1(1):79-97, March 1981.
28. D.L. Wang and D.J. Kleitman. On the existence of n-connected graphs with prescribed degrees (n > 2). Networks, 3:225-239, 1973.