Treewidth versus clique number. II. Tree-independence number

Dallard, Clément; Milanič, Martin; Štorgel, Kenny

doi:10.1016/j.jctb.2023.10.006

Cited by 4 publications

(10 citation statements)

References 65 publications

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Mentioning

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“…As a consequence of these results, we obtain that the classes

{MJX-tex-caligraphicscriptG}_{k}

(

k \geq 0

) are closed under induced minors and give a complete list of minimal forbidden induced minors for the class

{MJX-tex-caligraphicscriptG}_{2}

. Combining these results with some results from the literature [16, 22], we show that for every integer

k \geq 0

, the Maximum Weight Stable Set Problem can be solved in polynomial time for graphs in

{MJX-tex-caligraphicscriptG}_{k}

, and we further show that all 1‐perfectly‐orientable graphs belong to

{MJX-tex-caligraphicscriptG}_{2}

.…”

Section: Introductionsupporting

confidence: 64%

“…It was shown by Dallard et al in [16] that, for each positive integer

k

, the Maximum Weight Stable Set problem can be solved in

O (n^{2 k})

time for

n

‐vertex

K_{2, k}

‐induced‐minor‐free graphs. To connect this result with the classes

{MJX-tex-caligraphicscriptG}_{k}

, by Corollary 3.2 every graph in

{MJX-tex-caligraphicscriptG}_{k}

K_{2, k + 1}

‐induced‐minor‐free.…”

Section: Forbidden Induced Minorsmentioning

confidence: 99%

“…Furthermore, Dallard et al showed in [17] that in any class of

K_{2, k}

‐induced‐minor‐free graphs, the treewidth of the graphs in the class is bounded from above by some polynomial function of the clique number (see also [16]). Combining this with a result of Chaplick et al [11, theorem 12], it follows that for any two positive integers

k

and

ℓ

, the

ℓ

‐ Coloring problem is solvable in time

O (n)

in the class of

n

‐vertex

K_{2, k}

‐induced‐minor‐free graphs, and thus in the class

{MJX-tex-caligraphicscriptG}_{k}

as well.…”

Section: Forbidden Induced Minorsmentioning

confidence: 99%

“…First, the Vertex Coloring problem is NP‐hard for diamond‐free graphs [24], as well as for graphs in

{MJX-tex-caligraphicscriptG}_{2}

, since it is already hard for the subclass of circular‐arc graphs [21]. The situation is somewhat different for the Maximum Weight Stable Set problem, which is NP‐hard (even in the unweighted case) in the class of diamond‐free graphs, as can be seen using Poljak's reduction [26], but solvable in

O (n^{5})

time for

n

‐vertex graphs in

{MJX-tex-caligraphicscriptG}_{2}

[16, theorem 3.16].…”

Section: Diamond‐free Graphs In Mjx-tex-caligraphicscriptg2mentioning

confidence: 99%

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Bisimplicial separators

Milanič,

Penev,

Pivač

et al. 2024

Journal of Graph Theory

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A minimal separator of a graph is a set such that there exist vertices with the property that separates from in , but no proper subset of does. For an integer , we say that a minimal separator is ‐simplicial if it can be covered by cliques and denote by the class of all graphs in which each minimal separator is ‐simplicial. We show that for each , the class is closed under induced minors, and we use this to show that the Maximum Weight Stable Set problem can be solved in polynomial time for . We also give a complete list of minimal forbidden induced minors for . Next, we show that, for , every nonnull graph in has a ‐simplicial vertex, that is, a vertex whose neighborhood is a union of cliques; we deduce that the Maximum Weight Clique problem can be solved in polynomial time for graphs in . Further, we show that, for , it is NP‐hard to recognize graphs in ; the time complexity of recognizing graphs in is unknown. We also show that the Maximum Clique problem is NP‐hard for graphs in . Finally, we prove a decomposition theorem for diamond‐free graphs in (where the diamond is the graph obtained from by deleting one edge), and we use this theorem to obtain polynomial‐time algorithms for the Vertex Coloring and recognition problems for diamond‐free graphs in , and improved running times for the Maximum Weight Clique and Maximum Weight Stable Set problems for this class of graphs.

show abstract

“…As a consequence of these results, we obtain that the classes

{MJX-tex-caligraphicscriptG}_{k}

(

k \geq 0

) are closed under induced minors and give a complete list of minimal forbidden induced minors for the class

{MJX-tex-caligraphicscriptG}_{2}

. Combining these results with some results from the literature [16, 22], we show that for every integer

k \geq 0

, the Maximum Weight Stable Set Problem can be solved in polynomial time for graphs in

{MJX-tex-caligraphicscriptG}_{k}

, and we further show that all 1‐perfectly‐orientable graphs belong to

{MJX-tex-caligraphicscriptG}_{2}

.…”

Section: Introductionsupporting

confidence: 64%

“…It was shown by Dallard et al in [16] that, for each positive integer

k

, the Maximum Weight Stable Set problem can be solved in

O (n^{2 k})

time for

n

‐vertex

K_{2, k}

‐induced‐minor‐free graphs. To connect this result with the classes

{MJX-tex-caligraphicscriptG}_{k}

, by Corollary 3.2 every graph in

{MJX-tex-caligraphicscriptG}_{k}

K_{2, k + 1}

‐induced‐minor‐free.…”

Section: Forbidden Induced Minorsmentioning

confidence: 99%

“…Furthermore, Dallard et al showed in [17] that in any class of

K_{2, k}

k

and

ℓ

, the

ℓ

‐ Coloring problem is solvable in time

O (n)

in the class of

n

‐vertex

K_{2, k}

‐induced‐minor‐free graphs, and thus in the class

{MJX-tex-caligraphicscriptG}_{k}

as well.…”

Section: Forbidden Induced Minorsmentioning

confidence: 99%

“…First, the Vertex Coloring problem is NP‐hard for diamond‐free graphs [24], as well as for graphs in

{MJX-tex-caligraphicscriptG}_{2}

O (n^{5})

time for

n

‐vertex graphs in

{MJX-tex-caligraphicscriptG}_{2}

[16, theorem 3.16].…”

Section: Diamond‐free Graphs In Mjx-tex-caligraphicscriptg2mentioning

confidence: 99%

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Bisimplicial separators

Milanič,

Penev,

Pivač

et al. 2024

Journal of Graph Theory

Self Cite

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“…The tree independence number was defined and studied by Dallard, Milanič, and Štorgel in [16], in the context of studying the complexity of the Maximum Weight Independent Set (MWIS) problem on graph classes whose treewidth is large only due to the presence of a large clique. It is shown in [16] that if a graph is given together with a tree decomposition with bounded independence number, then the MWIS problem can be solved in polynomial time. In [17], it is then shown how to compute such tree decompositions efficiently in graphs of bounded

tree‐ α

, yielding an efficient algorithm for the MWIS problem for graphs of bounded

tree‐ α

.…”

Section: Introductionmentioning

confidence: 99%

Tree independence number I. (Even hole, diamond, pyramid)‐free graphs

Abrishami,

Alecu,

Chudnovsky

et al. 2024

Journal of Graph Theory

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The tree‐independence number , first defined and studied by Dallard, Milanič, and Štorgel, is a variant of treewidth tailored to solving the maximum independent set problem. Over a series of papers, Abrishami et al. developed the so‐called central bag method to study induced obstructions to bounded treewidth. Among others, they showed that, in a certain superclass of (even hole, diamond, pyramid)‐free graphs, treewidth is bounded by a function of the clique number. In this paper, we relax the bounded clique number assumption, and show that has bounded . Via existing results, this yields a polynomial‐time algorithm for the Maximum Weight Independent Set problem in this class. Our result also corroborates, for this class of graphs, a conjecture of Dallard, Milanič, and Štorgel that in a hereditary graph class, is bounded if and only if the treewidth is bounded by a function of the clique number.

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