Extending Partial Representations of Proper and Unit Interval Graphs

Klavík, Pavel; Kratochvı́l, Jan; Otachi, Yota; Rutter, Ignaz; Saitoh, Toshiki; Saumell, Maria; Vyskočil, T

doi:10.48550/arxiv.1207.6960

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“…Our results. While for many classes efficient algorithms for the representation extension problem have been found, the problem has been open for circular arc graphs for nine years [17]. We prove that RepExt(CAR) is NP-hard.…”

Section: Introductionmentioning

confidence: 98%

Extending Partial Representations of Circular-Arc Graphs

Fiala¹,

Rutter²,

Stumpf³

et al. 2021

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The partial representation extension problem generalizes the recognition problem for classes of graphs defined in terms of vertex representations. We exhibit circular-arc graphs as the first example of a graph class where the recognition is polynomially solvable while the representation extension problem is NP-complete. In this setting, several arcs are predrawn and we ask whether this partial representation can be completed.We complement this hardness argument with tractability results of the representation extension problem on various subclasses of circular-arc graphs, most notably on all variants of Helly circular-arc graphs. In particular, we give linear-time algorithms for extending normal proper Helly and proper Helly representations. For normal Helly circular-arc representations we give an O(n 3 )-time algorithm.Surprisingly, for Helly representations, the complexity hinges on the seemingly irrelevant detail of whether the predrawn arcs have distinct or non-distinct endpoints: In the former case the previous algorithm can be extended, whereas the latter case turns out to be NP-complete. We also prove that representation extension problem of unit circular-arc graphs is NP-complete.

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Section: Introductionmentioning

confidence: 98%