Abstract
Function graphs are graphs representable by intersections of continuous real-valued functions on the interval [0,1] and are known to be exactly the complements of comparability graphs. As such they are recognizable in polynomial time. Function graphs generalize permutation graphs, which arise when all functions considered are linear.
We focus on the problem of extending partial representations, which generalizes the recognition problem. We observe that for permutation graphs an easy extension of Golumbic’s comparability graph recognition algorithm can be exploited. This approach fails for function graphs. Nevertheless, we present a polynomial-time algorithm for extending a partial representation of a graph by functions defined on the entire interval [0,1] provided for some of the vertices. On the other hand, we show that if a partial representation consists of functions defined on subintervals of [0,1], then the problem of extending this representation to functions on the entire interval [0,1] becomes NP-complete.
Supported by ESF EuroGIGA project GraDR, the first two authors by Czech Science Foundation as grant No. GIG/11/E023, the last two authors by Ministry of Science and Higher Education of Poland as grant No. 884/N-ESF-EuroGIGA/10/2011/0.
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Klavík, P., Kratochvíl, J., Krawczyk, T., Walczak, B. (2012). Extending Partial Representations of Function Graphs and Permutation Graphs. In: Epstein, L., Ferragina, P. (eds) Algorithms – ESA 2012. ESA 2012. Lecture Notes in Computer Science, vol 7501. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33090-2_58
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DOI: https://doi.org/10.1007/978-3-642-33090-2_58
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