Abstract
We consider the problem of finding interval representations of graphs that additionally respect given interval lengths and/or pairwise intersection lengths, which are represented as weight functions on the vertices and edges, respectively. Pe’er and Shamir proved that the problem is \(\text{\upshape\textsf{NP}}\)-complete if only the former are given [SIAM J. Discr. Math. 10.4, 1997]. We give both a linear-time and a logspace algorithm for the case when both are given, and both an \(\ensuremath{\mathcal{O}}(n\cdot m)\) time and a logspace algorithm when only the latter are given. We also show that the resulting interval systems are unique up to isomorphism.
Complementing their hardness result, Pe’er and Shamir give a polynomial-time algorithm for the case that the input graph has a unique interval ordering of its maxcliques. For such graphs, their algorithm computes an interval representation that respects a given set of distance inequalities between the interval endpoints (if it exists). We observe that deciding if such a representation exists is \(\text{\upshape\textsf{NL}}\)-complete.
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Köbler, J., Kuhnert, S., Watanabe, O. (2012). Interval Graph Representation with Given Interval and Intersection Lengths. In: Chao, KM., Hsu, Ts., Lee, DT. (eds) Algorithms and Computation. ISAAC 2012. Lecture Notes in Computer Science, vol 7676. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35261-4_54
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DOI: https://doi.org/10.1007/978-3-642-35261-4_54
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