The chain graph sandwich problem

Dantas, Simone; Figueiredo, Celina M. H. de; Golumbic, Martin Charles; Klein, Sulamita; Maffray, Frédéric

doi:10.1007/s10479-010-0792-0

Cited by 9 publications

(8 citation statements)

References 14 publications

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“…The proof is in Appendix. The details of the algorithm are also shown in [7]. ⊓ ⊔ Then, we show that E r has a chain completion in G − F. Lemma 6.…”

Section: Adding Edgesmentioning

confidence: 94%

“…We next show that E r can be extended into a chain graph in G − F, the subgraph of G obtained by removing all the edges in F. To do this, we consider the following problem: Given a graph H and a set M of edges of H, find a chain subgraph C of H containing all edges in M. This problem is called the chain graph sandwich problem, and the chain graph C is called a chain completion of M in H. Although the chain graph sandwich problem is NP-complete, it can be solved in linear time if H is a bipartite graph [7]. The chain graph sandwich problem on bipartite graphs is closely related to the threshold graph sandwich problem [8,19] (see also Section 1.5 of [15]), and in the proof of Lemma 5, we will use an argument similar to that used in the literature.…”

Section: ⊓ ⊔mentioning

confidence: 99%

“…Let u be such a vertex, and we assume without loss of generality that u ∈ U. The proof of "if" part gives a linear-time algorithm that finds a chain completion of M in G. The details of the algorithm are also shown in [7].…”

mentioning

confidence: 99%

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Recognizing Simple-Triangle Graphs by Restricted 2-Chain Subgraph Cover

Takaoka¹

2017

WALCOM: Algorithms and Computation

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A simple-triangle graph (also known as a PI graph) is the intersection graph of a family of triangles defined by a point on a horizontal line and an interval on another horizontal line. The recognition problem for simple-triangle graphs was a longstanding open problem, and recently a polynomial-time algorithm has been given [G. B. Mertzios, The Recognition of Simple-Triangle Graphs and of Linear-Interval Orders is Polynomial, SIAM J. Discrete Math., 29(3):1150--1185, 2015]. Along with the approach of this paper, we show a simpler recognition algorithm for simple-triangle graphs. To do this, we provide a polynomial-time algorithm to solve the following problem: Given a bipartite graph $G$ and a set $F$ of edges of $G$, find a 2-chain subgraph cover of $G$ such that one of two chain subgraphs has no edges in $F$.Comment: 13 pages, 14 figures, the Author's accepted version of a paper in WALCOM 2017, Keywords: Chain cover, Graph sandwich problem, PI graphs, Simple-triangle graphs, Threshold dimension 2 graph

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“…The proof is in Appendix. The details of the algorithm are also shown in [7]. ⊓ ⊔ Then, we show that E r has a chain completion in G − F. Lemma 6.…”

Section: Adding Edgesmentioning

confidence: 94%

Section: ⊓ ⊔mentioning

confidence: 99%

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Recognizing Simple-Triangle Graphs by Restricted 2-Chain Subgraph Cover

Takaoka¹

2017

WALCOM: Algorithms and Computation

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“…Proof. We prove the lemma by showing an algorithm to construct an interval representation of P I from P and L. We note that this algorithm is inspired by the algorithms that solve the sandwich problems for chain graphs and for threshold graphs [7,10,14,18,21].…”

Section: Lemmamentioning

confidence: 99%

A vertex ordering characterization of simple-triangle graphs

Takaoka

2018

Discrete Mathematics

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Consider two horizontal lines in the plane. A pair of a point on the top line and an interval on the bottom line defines a triangle between two lines. The intersection graph of such triangles is called a simple-triangle graph. This paper shows a vertex ordering characterization of simple-triangle graphs as follows: a graph is a simple-triangle graph if and only if there is a linear ordering of the vertices that contains both an alternating orientation of the graph and a transitive orientation of the complement of the graph.

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“…Graph sandwich problems [12] are a natural generalization of recognition problems, and have received considerable attention [5,[7][8][9][10]13,18,19]. It is not unusual that graph classes for which the recognition is very easy lead to challenging graph sandwich problems, which are either intractable or require interesting structural and algorithmic arguments for their solution.…”

Section: Introductionmentioning

confidence: 99%

Sandwiches missing two ingredients of order four

2019

Self Cite

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For a set F of graphs, an instance of the F -free Sandwich Problem is a pair (G 1 , G 2 ) consisting of two graphs G 1 and G 2 with the same vertex set such that G 1 is a subgraph of G 2 , and the task is to determine an F -free graph G containing G 1 and contained in G 2 , or to decide that such a graph does not exist. Initially motivated by the graph sandwich problem for trivially perfect graphs, which are the {P 4 , C 4 }-free graphs, we study the complexity of the F -free Sandwich Problem for sets F containing two non-isomorphic graphs of order four. We show that if F is one of the sets {diamond,

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The chain graph sandwich problem

Cited by 9 publications

References 14 publications

Recognizing Simple-Triangle Graphs by Restricted 2-Chain Subgraph Cover

Recognizing Simple-Triangle Graphs by Restricted 2-Chain Subgraph Cover

A vertex ordering characterization of simple-triangle graphs

Sandwiches missing two ingredients of order four

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