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## Sparse graphs with bounded induced cycle packing number have logarithmic treewidth

Marthe Bonamy
Édouard Bonnet
• Function : Author
Hugues Déprés
• Function : Author
Louis Esperet
Colin Geniet
• Function : Author
Claire Hilaire
• Function : Author
Stéphan Thomassé
• Function : Author
Alexandra Wesolek
• Function : Author

#### Abstract

A graph is $O_k$-free if it does not contain $k$ pairwise vertex-disjoint and non-adjacent cycles. We prove that "sparse" (here, not containing large complete bipartite graphs as subgraphs) $O_k$-free graphs have treewidth (even, feedback vertex set number) at most logarithmic in the number of vertices, which is sharp already for $k=2$. As a consequence, most of the central NP-complete problems (such as Maximum Independent Set, Minimum Vertex Cover, Minimum Dominating Set, Minimum Coloring) can be solved in polynomial time in these graphs, and in particular deciding the $O_k$-freeness of sparse graphs is polytime.

### Dates and versions

hal-03685279 , version 1 (02-06-2022)

### Identifiers

• HAL Id : hal-03685279 , version 1
• ARXIV :

### Cite

Marthe Bonamy, Édouard Bonnet, Hugues Déprés, Louis Esperet, Colin Geniet, et al.. Sparse graphs with bounded induced cycle packing number have logarithmic treewidth. 2022. ⟨hal-03685279⟩

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