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Abstract A class of graphs admits the Erdős–Pósa property if for any graph , either has vertex‐disjoint “copies” of the graphs in , or there is a set of vertices that intersects all copies of the graphs in . For any graph class , it is natural to ask whether the family of obstructions to has the Erdős–Pósa property. In this paper, we prove that the family of obstructions to interval graphs—namely, the family of chordless cycles and asteroidal witnesses (AWs)—admits the Erdős–Pósa property. In turn, this yields an algorithm to decide whether a given graph has vertex‐disjoint AWs and chordless cycles, or there exists a set of vertices in that hits all AWs and chordless cycles.
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Acyclic graphs with at least 2ℓ + 1 vertices are ℓ‐recognizable
Abstract The ‐deckof an ‐vertex graph is the multiset of subgraphs obtained from it by deleting vertices. A family of ‐vertex graphs is ‐recognizableif every graph having the same ‐deck as a graph in the family is also in the family. We prove that the family of ‐vertex graphs with no cycles is ‐recognizable when (except for ). As a consequence, the family of ‐vertex trees is ‐recognizable when and . It is known that this fails when .
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- PAR ID:
- 10444556
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Journal of Graph Theory
- ISSN:
- 0364-9024
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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