Domination and Cut Problems on Chordal Graphs with Bounded Leafage

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.