Domination and Cut Problems on Chordal Graphs with Bounded Leafage

<jats:p>The leafage of a chordal graph <jats:italic>G</jats:italic> is the minimum integer <jats:inline-formula><jats:alternatives><jats:tex-math>$$\ell $$</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> such that <jats:italic>G</jats:italic> can be realized as an intersection graph of subtrees of a tree with <jats:inline-formula><jats:alternatives><jats:tex-math>$$\ell $$</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> 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 <jats:sc>Dominating Set</jats:sc> on chordal graphs admits an algorithm running in time <jats:inline-formula><jats:alternatives><jats:tex-math>$$2^{\mathcal {O}(\ell ^2)} \cdot n^{\mathcal {O}(1)}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> <mml:mo>·</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>. We present a conceptually much simpler algorithm that runs in time <jats:inline-formula><jats:alternatives><jats:tex-math>$$2^{\mathcal {O}(\ell )} \cdot n^{\mathcal {O}(1)}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> <mml:mo>·</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>. We extend our approach to obtain similar results for <jats:sc>Connected Dominating Set</jats:sc> and <jats:sc>Steiner Tree</jats:sc>. We then consider the two classical cut problems <jats:sc>MultiCut with Undeletable Terminals</jats:sc> and <jats:sc>Multiway Cut with Undeletable Terminals</jats:sc>. We prove that the former is [1]-hard when parameterized by the leafage and complement this result by presenting a simple <jats:inline-formula><jats:alternatives><jats:tex-math>$$n^{\mathcal {O}(\ell )}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML"> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula>-time algorithm. To our surprise, we find that <jats:sc>Multiway Cut with Undeletable Terminals</jats:sc> on chordal graphs can be solved, in contrast, in <jats:inline-formula><jats:alternatives><jats:tex-math>$$n^{{{\mathcal {O}}}(1)}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML"> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula>-time. </jats:p>