Extending partial representations of function graphs and permutation graphs

Authors:
Pavel Klavík;Jan Kratochvíl;Tomasz Krawczyk;Bartosz Walczak
Affiliations:
Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Czech Republic;Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Czech Republic;Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Poland;Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Poland
Venue:
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
Year:
2012

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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.