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Bera, Gishboliner, Levanzov, Seshadhri, and Shapira (SODA, 2021) showed that a pattern graph has DAG treewidth one if and only if it contains no induced cycle of length at least six. This induced minor characterization leads to linear-time and linear-space algorithms. Building on this line of work, we derive an induced-minor characterization of graphs with DAG treedepth at most two that uses only constant space.
Recently, Paul-Pena and Seshadhri (ICALP, 2025) proved that all pattern graphs on at most nine vertices can be counted in subquadratic time using polynomial space. We show that every pattern graph on at most nine vertices can be counted as an induced subgraph in $O(n^3)$ time using only constant space. Moreover, we show that patterns on at most eleven vertices can be counted in $O(n^2)$ time using polynomial space.
Finally, we present a constant-space algorithm for counting induced subgraphs that matches the running time of Bressan algorithm. We further show that, when polynomial space is allowed, homomorphisms, subgraph isomorphisms, and induced subgraph isomorphisms can be counted faster than Bressan algorithm. In addition, we establish several other results related to DAG treewidth and DAG treedepth that may be of independent interest.
From: Akash Pareek [view email]
[v1]
Thu, 6 Nov 2025 10:49:07 UTC (403 KB)
[v2]
Tue, 30 Jun 2026 13:57:37 UTC (359 KB)
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