Minimal Obstructions for Partial Representations of Interval Graphs

Klavík, Pavel; Saumell, Maria

doi:10.37236/5862

Cited by 2 publications

(2 citation statements)

References 31 publications

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“…For chordal graphs (as subtree-in-a-tree graphs) several versions of the problems were considered [27] and all of them are NP-complete, and similarly for different contact representations of planar graphs [9]. In [30], minimal forbidden configurations making a partial interval representation nonextendible are characterized. Extending partial visibility representations is studied in [11].…”

Section: Tionmentioning

confidence: 99%

Extending partial representations of circle graphs

Chaplick

Fulek

Klavík

2018

Journal of Graph Theory

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The partial representation extension problem is a recently introduced generalization of the recognition problem. A circle graph is an intersection graph of chords of a circle. We study the partial representation extension problem for circle graphs, where the input consists of a graph G and a partial representation R′ giving some predrawn chords that represent an induced subgraph of G. The question is whether one can extend R ′ to a representation scriptR of the entire graph G, that is, whether one can draw the remaining chords into a partially predrawn representation to obtain a representation of G. Our main result is an scriptO ( n 3 ) time algorithm for partial representation extension of circle graphs, where n is the number of vertices. To show this, we describe the structure of all representations of a circle graph using split decomposition. This can be of independent interest.

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Section: Tionmentioning

confidence: 99%

Extending partial representations of circle graphs

Chaplick

Fulek

Klavík

2018

Journal of Graph Theory

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“…There are two different linear-time algorithms [1,19] for this problem. Minimal obstructions making partial representations non-extendible are described in [23]. A linear-time algorithm for proper interval graphs [17] and a quadratic-time algorithm for unit interval graphs [31] are known.…”

Section: Problem: Partial Representation Extension -Repext(c)mentioning

confidence: 99%

On the Classes of Interval Graphs of Limited Nesting and Count of Lengths

2018

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In 1969, Roberts introduced proper and unit interval graphs and proved that these classes are equal. Natural generalizations of unit interval graphs called k-length interval graphs were considered in which the number of different lengths of intervals is limited by k. Even after decades of research, no insight into their structure is known and the complexity of recognition is open even for k = 2. We propose generalizations of proper interval graphs called k-nested interval graphs in which there are no chains of k + 1 intervals nested in each other. It is easy to see that k-nested interval graphs are a superclass of k-length interval graphs.We give a linear-time recognition algorithm for k-nested interval graphs. This algorithm adds a missing piece to Gajarský et al. [FOCS 2015] to show that testing FO properties on interval graphs is FPT with respect to the nesting k and the length of the formula, while the problem is W[2]-hard when parameterized just by the length of the formula. We show that a generalization of recognition called partial representation extension is NP-hard for k-length interval graphs, even when k = 2, while Klavík et al. show that it is polynomial-time solvable for k-nested interval graphs.

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Minimal Obstructions for Partial Representations of Interval Graphs

Cited by 2 publications

References 31 publications

Extending partial representations of circle graphs

Extending partial representations of circle graphs

On the Classes of Interval Graphs of Limited Nesting and Count of Lengths

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