Abstract
We study Subgraph Isomorphism on graph classes defined by a fixed forbidden graph. Although there are several ways for forbidding a graph, we observe that it is reasonable to focus on the minor relation since other well-known relations lead to either trivial or equivalent problems. When the forbidden minor is connected, we present a near dichotomy of the complexity of Subgraph Isomorphism with respect to the forbidden minor, where the only unsettled case is P-5, the path of five vertices. We then also consider the general case of possibly disconnected forbidden minors. We show fixed-parameter tractable cases and randomized XP-time solvable cases parameterized by the size of the forbidden minor H. We also show that by slightly generalizing the tractable cases, the problem becomes NP-complete. All unsettle cases are equivalent to P-5 or the disjoint union of two P-5's. As a byproduct, we show that Subgraph Isomorphism is fixed-parameter tractable parameterized by vertex integrity. Using similar techniques, we also observe that Subgraph Isomorphism is fixed-parameter tractable parameterized by neighborhood diversity.
Original language | English |
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Pages (from-to) | 3566-3587 |
Number of pages | 22 |
Journal | Algorithmica |
Volume | 82 |
Issue number | 12 |
Early online date | 2 Jul 2020 |
DOIs | |
Publication status | Published - Dec 2020 |
Keywords
- Subgraph isomorphism
- Minor-free graphs
- Parameterized complexity
- MODULAR DECOMPOSITION
- COMPLEXITY