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Characterization and Recognition of Proper Tagged Probe Interval Graphs

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Intelligent Computing (CompCom 2019)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 998))

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Abstract

Interval graphs were used in the study of genomics by the famous molecular biologist Benzer. Later on probe interval graphs were introduced by Zhang as a generalization of interval graphs for the study of cosmid contig mapping of DNA. A tagged probe interval graph (briefly, TPIG) is motivated by similar applications to genomics, where the set of vertices is partitioned into two sets, namely, probes and nonprobes and there is an interval on the real line corresponding to each vertex. The graph has an edge between two probe vertices if their corresponding intervals intersect, has an edge between a probe vertex and a nonprobe vertex if the interval corresponding to a nonprobe vertex contains at least one end point of the interval corresponding to a probe vertex and the set of nonprobe vertices is an independent set. This class of graphs have been defined nearly two decades ago, but till today there is no known recognition algorithm for it. In this paper, we consider a natural subclass of TPIG, namely, the class of proper tagged probe interval graphs (in short PTPIG). We present characterization and a linear time recognition algorithm for PTPIG. To obtain this characterization theorem we introduce a new concept called canonical sequence for proper interval graphs, which, we belief, has an independent interest in the study of proper interval graphs. Also to obtain the recognition algorithm for PTPIG, we introduce and solve a variation of consecutive 1’s problem, namely, oriented-consecutive 1’s problem and some variations of PQ-tree algorithm.

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Notes

  1. 1.

    In [27], Sheng et al. used the term “contains properly” in this case. Here we consider a different term in order to avoid confusion with the definition of proper interval graph. Note that if \(a\leqslant c\leqslant d<b\) or \(a<c\leqslant d\leqslant b\), then also \(I_p\) contains \(I_n\) properly, but not strongly.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Jadavpur University, Kolkata, 700032, India

    Sanchita Paul

  2. Department of Mathematics, Jadavpur University, Kolkata, 700032, India

    Shamik Ghosh

  3. Indian Statistical Institute, Kolkata, 700108, India

    Sourav Chakraborty

  4. Department of Mathematics, North Bengal University, Siliguri, 734430, West Bengal, India

    Malay Sen

Authors
  1. Sanchita Paul
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  2. Shamik Ghosh
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  3. Sourav Chakraborty
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  4. Malay Sen
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Corresponding author

Correspondence to Shamik Ghosh .

Editor information

Editors and Affiliations

  1. Faculty of Science and Engineering, Saga University, Saga, Japan

    Kohei Arai

  2. The Science and Information (SAI) Organization, Bradford, West Yorkshire, UK

    Rahul Bhatia

  3. The Science and Information (SAI) Organization, Bradford, West Yorkshire, UK

    Supriya Kapoor

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Paul, S., Ghosh, S., Chakraborty, S., Sen, M. (2019). Characterization and Recognition of Proper Tagged Probe Interval Graphs. In: Arai, K., Bhatia, R., Kapoor, S. (eds) Intelligent Computing. CompCom 2019. Advances in Intelligent Systems and Computing, vol 998. Springer, Cham. https://doi.org/10.1007/978-3-030-22868-2_5

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