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Directed acyclic outerplanar graphs have bounded stack number, resolving a long-standing conjecture.
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The key insights and highlights of the content are:
The stack number of a directed acyclic graph (DAG) G is the minimum number of "stacks" required to partition the edges of G such that no two edges of the same stack cross each other in the topological ordering of the vertices.
The authors prove that the stack number of directed acyclic outerplanar graphs is bounded by a constant, resolving a conjecture by Heath, Pemmaraju and Trenk from 1999. This also implies that all upward outerplanar graphs have constant stack number.
As a complementary result, the authors construct a family of directed acyclic 2-trees that have unbounded stack number, refuting a conjecture by Nöllenburg and Pupyrev.
The authors introduce a novel technique called "directed H-partitions" as a key tool to prove the bounded stack number for outerplanar DAGs. This technique may be of independent interest.
The proof proceeds by partitioning the outerplanar DAG into "transitive parts" such that the contraction of each part into a single vertex yields a block-monotone DAG. The stack layouts of the individual blocks can then be combined into a single stack layout of the overall graph.
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