$${\mathcal {U}}$$-Bubble Model for Mixed Unit Interval Graphs and Its Applications: The MaxCut Problem Revisited

Kratochvı́l, Jan; Masařík, Tomáš; Novotná, Jana

doi:10.1007/s00453-021-00837-4

Cited by 6 publications

(4 citation statements)

References 33 publications

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“…It is known that Max-Cut is NP-complete on cobipartite graphs [17], and thus on cocomparability graphs. Interestingly, the complexity of Max-Cut is still open on proper interval graphs [1,16,29,40]. We show that the problem remains NP-complete on cobipartite graphs with only two moplexes.…”

Section: Max-cutmentioning

confidence: 94%

Graphs with at most two moplexes

Dallard¹,

Ganian²,

Hatzel³

et al. 2021

Preprint

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Section: Max-cutmentioning

confidence: 94%

Graphs with at most two moplexes

Dallard¹,

Ganian²,

Hatzel³

et al. 2021

Preprint

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show abstract

“…It is known that MAX-CUT is NP-complete on cobipartite graphs [17], and thus on cocomparability graphs. Interestingly, the complexity of MAX-CUT is still open on proper interval graphs [1,16,30,41].…”

Section: Max-cutmentioning

confidence: 99%

Graphs with at most two moplexes

Dallard,

Ganian,

Hatzel

et al. 2024

Journal of Graph Theory

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A moplex is a natural graph structure that arises when lifting Dirac's classical theorem from chordal graphs to general graphs. The notion is known to be closely related to lexicographic searches in graphs as well as to asteroidal triples, and has been applied in several algorithms related to graph classes, such as interval graphs, claw‐free, and diamond‐free graphs. However, while every noncomplete graph has at least two moplexes, little is known about structural properties of graphs with a bounded number of moplexes. The study of these graphs is, in part, motivated by the parallel between moplexes in general graphs and simplicial modules in chordal graphs: unlike in the moplex setting, properties of chordal graphs with a bounded number of simplicial modules are well understood. For instance, chordal graphs having at most two simplicial modules are intervals. In this work, we initiate an investigation of ‐moplex graphs, which are defined as graphs containing at most moplexes. Of particular interest is the smallest nontrivial case , which forms a counterpart to the class of interval graphs. As our main structural result, we show that, when restricted to connected graphs, the class of 2‐moplex graphs is sandwiched between the classes of proper interval graphs and cocomparability graphs; moreover, both inclusions are tight for hereditary classes. From a complexity‐theoretic viewpoint, this leads to the natural question of whether the presence of at most two moplexes guarantees a sufficient amount of structure to efficiently solve problems that are known to be intractable on cocomparability graphs, but not on proper interval graphs. We develop new reductions that answer this question negatively for two prominent problems fitting this profile, namely, Graph Isomorphism and Max‐Cut. On the other hand, we prove that every connected 2‐moplex graph contains a Hamiltonian path, generalising the same property of connected proper interval graphs. Furthermore, for graphs with a higher number of moplexes, we lift the previously known result that graphs without asteroidal triples have at most two moplexes to the more general setting of larger asteroidal sets.

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“…interval graphs of interval count 1, still remains unknown. In fact, many flawed proofs of polynomial-time solvability for the problem on the class of unit interval graphs have been presented [Bodlaender et al 1999, Boyaci et al 2017], just to be disproved closely after [Bodlaender et al 2004, Kratochvíl et al 2020. We present the first complexity result for MAXCUT on interval graphs of bounded interval count, by proving that the problem remains NP-complete on interval graphs of interval count 4.…”

Section: Introductionmentioning

confidence: 99%

On (in)tractability of connection and cut problems

Melo

Figueiredo

Souza³

et al. 2023

Anais Do XXXVI Concurso De Teses E Dissertações (CTD 2023)

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This work addresses connection and cut problems from the viewpoint of graph classes and computational complexity, classic and parameterized. Regarding connection problems, we investigate the so-called TERMINAL CONNECTION problem (TCP), which can be seen as a generalisation of the classical STEINER TREE problem. We propose several complexity results for TCP, when restricted to specific graph classes, and some of its input parameters are fixed. As for cut problems, we analyse the complexity of the classical MAXCUT problem. We introduce the first complexity classification for the problem on interval graphs of bounded interval count. In addition, we prove the NP-completeness of MAXCUT on permutation graphs, settling a question posed by David S. Johnson in the Ongoing Guide to NP-completeness, which has been open since 1985. Finally, we consider the problem of computing the zig-zag number of a directed graph, which is a directed width measure defined over cuts of a graph.

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$${\mathcal {U}}$$-Bubble Model for Mixed Unit Interval Graphs and Its Applications: The MaxCut Problem Revisited

Cited by 6 publications

References 33 publications

Graphs with at most two moplexes

Graphs with at most two moplexes

Graphs with at most two moplexes

On (in)tractability of connection and cut problems

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