An $$O(\log \mathrm {OPT})$$ O ( log OPT ) -Approximation for Covering and Packing Minor Models of $$\theta _r$$ θ r

Chatzidimitriou, Dimitrios; Raymond, Jean-Florent; Sau, Ignasi; Thilikos, Dimitrios M.

doi:10.1007/s00453-017-0313-5

Cited by 8 publications

(3 citation statements)

References 34 publications

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“…This was recently extended to the case where H is an arbitrary cycle [7] (see also [1,14] for related results). The conjecture also holds when H is a multigraph consisting of two vertices linked by a number of parallel edges [2].…”

Section: Introductionmentioning

confidence: 87%

A Tight Erdös--Pósa Function for Wheel Minors

Aboulker¹,

Fiorini²,

Huynh³

et al. 2018

SIAM J. Discrete Math.

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Let W t denote the wheel on t + 1 vertices. We prove that for every integer t ≥ 3 there is a constant c = c(t) such that for every integer k ≥ 1 and every graph G, either G has k vertex-disjoint subgraphs each containing W t as a minor, or there is a subset X of at most ck log k vertices such that G − X has no W t minor. This is best possible, up to the value of c. We conjecture that the result remains true more generally if we replace W t with any fixed planar graph H.

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Section: Introductionmentioning

confidence: 87%

A Tight Erdös--Pósa Function for Wheel Minors

Aboulker¹,

Fiorini²,

Huynh³

et al. 2018

SIAM J. Discrete Math.

Self Cite

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“…A tree-partition is a T-partition for some tree T. The tree-partition-width 3 of G, denoted by tpw(G), is the minimum width of a tree-partition of G. Thus tpw(G) = tpw 1 (G), which equals the minimum ∈ N 0 for which G is contained in T K for some tree T. Tree-partitions were independently introduced by Seese [69] and Halin [39], and have since been widely investigated [7,8,19,20,24,34,76,77]. Applications of tree-partitions include graph drawing [13,16,30,32,80], nonrepetitive graph colouring [2], clustered graph colouring [1], monadic second-order logic [51], network emulations [4,5,9,37], size Ramsey number [26,43], statistical learning theory [81], and the edge-Erdős-Pósa property [14,38,60]. Planar-partitions and other more general structures have also been studied [18,21,22,63,80].…”

Section: Partitionsmentioning

confidence: 99%

Product structure of graph classes with bounded treewidth

Campbell,

Clinch,

Distel

et al. 2023

Combinator. Probab. Comp.

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We show that many graphs with bounded treewidth can be described as subgraphs of the strong product of a graph with smaller treewidth and a bounded-size complete graph. To this end, define the underlying treewidth of a graph class $\mathcal{G}$ to be the minimum non-negative integer $c$ such that, for some function $f$ , for every graph $G \in \mathcal{G}$ there is a graph $H$ with $\textrm{tw}(H) \leqslant c$ such that $G$ is isomorphic to a subgraph of $H \boxtimes K_{f(\textrm{tw}(G))}$ . We introduce disjointed coverings of graphs and show they determine the underlying treewidth of any graph class. Using this result, we prove that the class of planar graphs has underlying treewidth $3$ ; the class of $K_{s,t}$ -minor-free graphs has underlying treewidth $s$ (for $t \geqslant \max \{s,3\}$ ); and the class of $K_t$ -minor-free graphs has underlying treewidth $t-2$ . In general, we prove that a monotone class has bounded underlying treewidth if and only if it excludes some fixed topological minor. We also study the underlying treewidth of graph classes defined by an excluded subgraph or excluded induced subgraph. We show that the class of graphs with no $H$ subgraph has bounded underlying treewidth if and only if every component of $H$ is a subdivided star, and that the class of graphs with no induced $H$ subgraph has bounded underlying treewidth if and only if every component of $H$ is a star.

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“…Tree-partitions were independently introduced by Seese [68] and Halin [38], and have since been widely investigated [7,8,19,20,31,75,76]. Applications of tree-partitions include graph drawing [12,15,28,30,79], nonrepetitive graph colouring [2], clustered graph colouring [1], monadic second-order logic [50], network emulations [4,5,9,36], statistical learning theory [80], and the edge-Erdős-Pósa property [13,37,59]. Planar-partitions and other more general structures have also been studied [17,21,22,62,79].…”

Section: Partitionsmentioning

confidence: 99%

Product structure of graph classes with bounded treewidth

Campbell¹,

Clinch²,

Marc³

et al. 2022

Preprint

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We show that many graphs with bounded treewidth can be described as subgraphs of the strong product of a graph with smaller treewidth and a bounded-size complete graph. To this end, define the underlying treewidth of a graph class G to be the minimum non-negative integer c such that, for some function f , for every graph G ∈ G there is a graphWe introduce disjointed coverings of graphs and show they determine the underlying treewidth of any graph class. Using this result, we prove that the class of planar graphs has underlying treewidth 3; the class of K s,t -minor-free graphs has underlying treewidth s (for t max{s, 3}); and the class of K t -minor-free graphs has underlying treewidth t − 2. In general, we prove that a monotone class has bounded underlying treewidth if and only if it excludes some fixed topological minor. We also study the underlying treewidth of graph classes defined by an excluded subgraph or excluded induced subgraph. We show that the class of graphs with no H subgraph has bounded underlying treewidth if and only if every component of H is a subdivided star, and that the class of graphs with no induced H subgraph has bounded underlying treewidth if and only if every component of H is a star.

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An $$O(\log \mathrm {OPT})$$ O ( log OPT ) -Approximation for Covering and Packing Minor Models of $$\theta _r$$ θ r

Cited by 8 publications

References 34 publications

A Tight Erdös--Pósa Function for Wheel Minors

A Tight Erdös--Pósa Function for Wheel Minors

Product structure of graph classes with bounded treewidth

Product structure of graph classes with bounded treewidth

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