Recognition of graphs with threshold dimension two

Raschle, Thomas; Simon, Klaus

doi:10.1145/225058.225283

Cited by 18 publications

(46 citation statements)

References 15 publications

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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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Section: Lemmamentioning

confidence: 99%

A vertex ordering characterization of simple-triangle graphs

Takaoka

2018

Discrete Mathematics

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“…This approach is used in the fastest known algorithm [14] with a running time of O(n 2 ), where n is the number of vertices of the given graph. Another approach can be found in [10,11,19]. They show that a bipartite graph G = (U, V, E) has a 2-chain subgraph cover if and only if the conflict graph G * = (V * , E * ) of G is bipartite, where V * = E and two edges e and e ′ in E are adjacent in G * if e and e ′ are in conflict in G. We note that the algorithm in this paper is based on the latter approach.…”

Section: Related Workmentioning

confidence: 99%

“…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: Adding Edgesmentioning

confidence: 99%

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 of the next lemma is a based on the results of [13]. Compute a 2-coloring χ0 of the vertices of H * 10:…”

Section: The Recognition Of Linear-interval Orders and Pi Graphsmentioning

confidence: 99%

The Recognition of Simple-Triangle Graphs and of Linear-Interval Orders Is Polynomial

Mertzios

2013

Lecture Notes in Computer Science

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Abstract. Intersection graphs of geometric objects have been extensively studied, both due to their interesting structure and their numerous applications; prominent examples include interval graphs and permutation graphs. In this paper we study a natural graph class that generalizes both interval and permutation graphs, namely simple-triangle graphs. Simple-triangle graphs -also known as PI graphs (for Point-Interval) -are the intersection graphs of triangles that are defined by a point on a line L1 and an interval on a parallel line L2. They lie naturally between permutation and trapezoid graphs, which are the intersection graphs of line segments between L1 and L2 and of trapezoids between L1 and L2, respectively. Although various efficient recognition algorithms for permutation and trapezoid graphs are well known to exist, the recognition of simple-triangle graphs has remained an open problem since their introduction by Corneil and Kamula three decades ago. In this paper we resolve this problem by proving that simple-triangle graphs can be recognized in polynomial time. As a consequence, our algorithm also solves a longstanding open problem in the area of partial orders, namely the recognition of linear-interval orders, i.e. of partial orders P = P1 ∩ P2, where P1 is a linear order and P2 is an interval order. This is one of the first results on recognizing partial orders P that are the intersection of orders from two different classes P1 and P2. In contrast, partial orders P which are the intersection of orders from the same class P have been extensively investigated, and in most cases the complexity status of these recognition problems has been established.

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Recognition of graphs with threshold dimension two

Cited by 18 publications

References 15 publications

A vertex ordering characterization of simple-triangle graphs

A vertex ordering characterization of simple-triangle graphs

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

The Recognition of Simple-Triangle Graphs and of Linear-Interval Orders Is Polynomial

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