Treewidth versus clique number. III. Tree-independence number of graphs with a forbidden structure

Dallard, Clément; Milanič, Martin; Štorgel, Kenny

doi:10.1016/j.jctb.2024.03.005

Cited by 2 publications

(2 citation statements)

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“…As a consequence of these results, we obtain that the classes  k ( ≥ k 0) are closed under induced minors and give a complete list of minimal forbidden induced minors for the class  2 . Combining these results with some results from the literature [16,22], we show that for every integer ≥ k 0, the MAXIMUM WEIGHT STABLE SET PROBLEM can be solved in polynomial time for graphs in  k , and we further show that all 1-perfectly-orientable graphs belong to  2 .…”

Section: Milanič Et Alsupporting

confidence: 70%

“…Furthermore, Dallard et al showed in [17] that in any class of K k 2, -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  n ( ) in the class of n-vertex K k 2, -induced-minor-free graphs, and thus in the class  k as well.…”

Section: Algorithmic Considerationsmentioning

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.

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Section: Milanič Et Alsupporting

confidence: 70%

Section: Algorithmic Considerationsmentioning

confidence: 99%