2026/07/30 by Kevin Hendrey, Robert Hickingbotham, Jędrzej Hodor +1
Mathematics · Computer Science · #math.CO #cs.DM
arxiv created 2026/07/30 · arxiv updated 2026/07/31
We show that every planar graph has a tree-decomposition with optimal width such that the subgraph induced by each bag has pathwidth at most 3. This bound is best possible, and for tree-decompositions that satisfy a certain minimality condition, we in fact give a precise description of the possible structures in each bag. Moreover, we show that the union of any k bags has pathwidth O(k). We also show that graphs excluding a fixed double-apex-forest minor have a tree-decomposition with optimal width such that the subgraph induced by each bag has bounded pathwidth. This includes graphs embeddable on any fixed surface. As a byproduct of our machinery, we give a new proof of the linear grid minor theorem for planar graphs.