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arXiv · 2004.07492

Steiner Trees for Hereditary Graph Classes: a Treewidth Perspective

Hans Bodlaender·Nick Brettell·Matthew Johnson·Giacomo Paesani·Daniel Paulusma·Erik Jan van Leeuwen
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Abstract

We consider the classical problems (Edge) Steiner Tree and Vertex Steiner Tree after restricting the input to some class of graphs characterized by a small set of forbidden induced subgraphs. We show a dichotomy for the former problem restricted to $(H_1,H_2)$-free graphs and a dichotomy for the latter problem restricted to $H$-free graphs. We find that there exists an infinite family of graphs $H$ such that Vertex Steiner Tree is polynomial-time solvable for $H$-free graphs, whereas there exist only two graphs $H$ for which this holds for Edge Steiner Tree. We also find that Edge Steiner Tree is polynomial-time solvable for $(H_1,H_2)$-free graphs if and only if the treewidth of the class of $(H_1,H_2)$-free graphs is bounded (subject to P $\neq$ NP). To obtain the latter result, we determine all pairs $(H_1,H_2)$ for which the class of $(H_1,H_2)$-free graphs has bounded treewidth.

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Hans Bodlaender, Nick Brettell, Matthew Johnson, Giacomo Paesani, Daniel Paulusma, Erik Jan van Leeuwen. 2020-04-16. Steiner Trees for Hereditary Graph Classes: a Treewidth Perspective. https://arxiv.org/abs/2004.07492

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