The firebreak problem

Barnetson, Kathleen D.; Burgess, Andrea C.; Enright, Jessica; Howell, Jared; Pike, David A.; Ryan, Brady

doi:10.1002/net.21975

Cited by 3 publications

(1 citation statement)

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“…Fomin et al [19] and Arvind et al [2] proved, respectively, that the Dominating Set and Graph Isomorphism problems on chordal graphs are FPT parameterized by the leafage. Barnetson et al [4] and Papadopoulos and Tzimas [44] presented XP-algorithms running in time n O(ℓ) for Fire Break and Subset FVS on chordal graphs, respectively. Papadopoulos and Tzimas [44] also proved that Subset FVS is W [1]hard when parameterized by the leafage.…”

Section: Introductionmentioning

confidence: 99%

Domination and Cut Problems on Chordal Graphs with Bounded Leafage

Galby¹,

Marx²,

Schepper³

et al. 2022

Preprint

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The leafage of a chordal graph G is the minimum integer ℓ such that G can be realized as an intersection graph of subtrees of a tree with ℓ leaves. We consider structural parameterization by the leafage of classical domination and cut problems on chordal graphs. Fomin, Golovach, and Raymond [ESA 2018, Algorithmica 2020] proved, among other things, that Dominating Set on chordal graphs admits an algorithm running in time 2 O(ℓ 2 ) • n O(1) . We present a conceptually much simpler algorithm that runs in time 2 O(ℓ) • n O(1) . We extend our approach to obtain similar results for Connected Dominating Set and Steiner Tree. We then consider the two classical cut problems MultiCut with Undeletable Terminals and Multiway Cut with Undeletable Terminals. We prove that the former is W[1]-hard when parameterized by the leafage and complement this result by presenting a simple n O(ℓ) -time algorithm. To our surprise, we find that Multiway Cut with Undeletable Terminals on chordal graphs can be solved, in contrast, in n O(1) -time.

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Section: Introductionmentioning

confidence: 99%

Domination and Cut Problems on Chordal Graphs with Bounded Leafage

Galby¹,

Marx²,

Schepper³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

Domination and Cut Problems on Chordal Graphs with Bounded Leafage

Galby,

Marx,

Schepper

et al. 2023

Algorithmica

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The leafage of a chordal graph G is the minimum integer $$\ell $$ ℓ such that G can be realized as an intersection graph of subtrees of a tree with $$\ell $$ ℓ leaves. We consider structural parameterization by the leafage of classical domination and cut problems on chordal graphs. Fomin, Golovach, and Raymond [ESA 2018, Algorithmica 2020] proved, among other things, that Dominating Set on chordal graphs admits an algorithm running in time $$2^{\mathcal {O}(\ell ^2)} \cdot n^{\mathcal {O}(1)}$$ 2 O ( ℓ 2 ) · n O ( 1 ) . We present a conceptually much simpler algorithm that runs in time $$2^{\mathcal {O}(\ell )} \cdot n^{\mathcal {O}(1)}$$ 2 O ( ℓ ) · n O ( 1 ) . We extend our approach to obtain similar results for Connected Dominating Set and Steiner Tree. We then consider the two classical cut problems MultiCut with Undeletable Terminals and Multiway Cut with Undeletable Terminals. We prove that the former is [1]-hard when parameterized by the leafage and complement this result by presenting a simple $$n^{\mathcal {O}(\ell )}$$ n O ( ℓ ) -time algorithm. To our surprise, we find that Multiway Cut with Undeletable Terminals on chordal graphs can be solved, in contrast, in $$n^{{{\mathcal {O}}}(1)}$$ n O ( 1 ) -time.

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