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

Klavík, Pavel; Otachi, Yota; Šejnoha, Jiří

doi:10.1007/s00453-018-0481-y

Cited by 9 publications

(8 citation statements)

References 33 publications

(60 reference statements)

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“…A k-nested interval graph is an interval graph admitting an interval representation in which there are no chains of k + 1 intervals nested in each other [24]. It is easy to see that k-nested interval graphs are a superclass of k-length interval graphs.…”

Section: Thus Pthin(clawmentioning

confidence: 99%

On the thinness and proper thinness of a graph

Bonomo

Estrada

2019

Discrete Applied Mathematics

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Graphs with bounded thinness were defined in 2007 as a generalization of interval graphs. In this paper we introduce the concept of proper thinness, such that graphs with bounded proper thinness generalize proper interval graphs. We study the complexity of problems related to the computation of these parameters, describe the behavior of the thinness and proper thinness under three graph operations, and relate thinness and proper thinness to other graph invariants in the literature. Finally, we describe a wide family of problems that can be solved in polynomial time for graphs with bounded thinness, generalizing for example list matrix partition problems with bounded size matrix, and enlarge this family of problems for graphs with bounded proper thinness, including domination problems.

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Section: Thus Pthin(clawmentioning

confidence: 99%

On the thinness and proper thinness of a graph

Bonomo

Estrada

2019

Discrete Applied Mathematics

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“…A linear-time algorithm for proper interval graphs and an almost quadratic-time algorithm for unit interval graphs are given in [19], and improved to quadratic time in [33]. The partial representation extension problems are polynomial-time solvable for k-nested interval graphs (classes generalizing proper interval graphs), but NP-hard for k-length interval graphs (classes generalizing unit interval graphs), even for k = 2 [23]. Polynomial-time algorithms are further known for circle graphs [7], and permutation and function graphs [18].…”

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

confidence: 99%

Minimal Obstructions for Partial Representations of Interval Graphs

Klavík

Saumell

2018

Electron. J. Combin.

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Interval graphs are intersection graphs of closed intervals. A generalization of recognition called partial representation extension was introduced recently. The input gives an interval graph with a partial representation specifying some pre-drawn intervals. We ask whether the remaining intervals can be added to create an extending representation. Two linear-time algorithms are known for solving this problem.In this paper, we characterize the minimal obstructions which make partial representations nonextendible. This generalizes Lekkerkerker and Boland's characterization of the minimal forbidden induced subgraphs of interval graphs. Each minimal obstruction consists of a forbidden induced subgraph together with at most four pre-drawn intervals. A Helly-type result follows: A partial representation is extendible if and only if every quadruple of pre-drawn intervals is extendible by itself. Our characterization leads to a linear-time certifying algorithm for partial representation extension. * The conference version of this paper appeared in ISAAC 2014 [24]. For an interactive structural diagram of this paper, see

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“…O tamanho de um intervalo I v , denotado por |I v |, é dado por r(I v ) − (I v ). O menor número IC(G) de tamanhos distintos de intervalos suficiente para se representar um modelo de G é chamado contagem de intervalo de G. Formalmente, IC(G) = min{IC(M) : M é um modelo de intervalo do grafo G} onde IC(M) denota o número de tamanhos distintos dos intervalos de M. O problema da contagem de intervalo foi inicialmente proposto por Ronald Graham na década de 80 e, até o momento, a complexidade de reconhecer se IC(G) ≤ k para todo k ≥ 2 é um problema em aberto [Fishburn 1985, Klavík et al 2019.…”

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