Abstract
Chordal graphs are intersection graphs of subtrees of a tree, while interval graphs are intersection graphs of subpaths of a path. Undirected path graphs are an intermediate class of graphs, defined as the intersection graphs of paths of a tree. It is known that Dominating Set, Connected Dominating Set, and Steiner Tree are \(\mathsf {W}[2]\)-hard on chordal graphs, when parameterized by the size of the solution, and are polynomial-time solvable on interval graphs. As for the undirected path graphs, all these problems are known to be \(\mathsf {NP}\)-complete, and when parameterized by the size of the solution, no classification in the parameterized complexity theory is known apart from the trivial \(\mathsf {XP}\) classification. We prove that Dominating Set, Connected Dominating Set, and Steiner Tree are \(\mathsf {FPT}\) for undirected path graphs when parameterized by the size of the solution, and that they continue to be \(\mathsf {FPT}\) for general chordal graphs when parameterized by the size of the solution plus the vertex leafage of the graph, provided a tree model with optimal vertex leafage is given. We show a relation between the parameterization of Min-LC-VSP problems by the leafage of the graph versus the vertex leafage plus the size of a solution.
This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001, CNPq grants 140399/2017-8, 407635/2018-1, and 303803/2020-7, FAPERJ grant E-26/202.793/2017, STIC-AMSUD 88881.197438/2018-01, and FUNCAP/CNPq PNE-0112-00061.01.00/16.
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de Figueiredo, C.M.H., Lopes, R., de Melo, A.A., Silva, A. (2022). Parameterized Algorithms for Steiner Tree and Dominating Set: Bounding the Leafage by the Vertex Leafage. In: Mutzel, P., Rahman, M.S., Slamin (eds) WALCOM: Algorithms and Computation. WALCOM 2022. Lecture Notes in Computer Science(), vol 13174. Springer, Cham. https://doi.org/10.1007/978-3-030-96731-4_21
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