Bipartite complements of circle graphs

Stehlı́k, Matěj

doi:10.1016/j.disc.2020.111834

Cited by 1 publication

(2 citation statements)

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“…Proof. In [8], the authors give an alternative proof of a result by Bouchet [1]. This result states that every graph that is a bipartite complement of a circle graph is also a circle graph.…”

Section: Subclasses Of Stick Graphsmentioning

confidence: 99%

“…This result states that every graph that is a bipartite complement of a circle graph is also a circle graph. Given a graph G = (A ∪ B, E) which is a bipartite complement of a circle graph, the proof of Theorem 1 in [8] shows that G is the intersection graph of a set of chords with pairwise distinct endpoints in a circle C and such that all the chords intersect a line l. The line l may be assumed horizontal such that p is the leftmost point of the circle and q its rightmost point. Assume G is connected (otherwise make the same reasoning for each connected component) and let S A , S B be the sets of chords corresponding to the vertices in A and B.…”

Section: Subclasses Of Stick Graphsmentioning

confidence: 99%

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Stick graphs: examples and counter-examples

Rusu¹

2020

Preprint

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Stick graphs are the intersection graphs of vertical and horizontal segments in the plane whose bottom and respectively left endpoints belong to a line with negative slope. Such a representation is called a Stick representation of the graph. Very few results exist on Stick graphs: only trivial classes of Stick graphs have been identified; recognizing Stick graphs is an open problem; and even building examples of graphs that are not Stick graphs is quite tricky. In this paper, we first propose structural results on Stick graphs, connecting the neighborhoods of two vertices to the order of these two vertices in a possible Stick representation. We deduce local obstructions for the existence of a Stick representation, which we use for the construction of graphs that are not Stick graphs. We next disapprove the idea, suggested by these examples and the other examples from the literature, that a graph which is not a Stick graph necessarily contains a hole. Finally, we show that bipartite complements of circulararc graphs and of circle graphs are subclasses of Stick graphs.

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Section: Subclasses Of Stick Graphsmentioning

confidence: 99%

Section: Subclasses Of Stick Graphsmentioning

confidence: 99%