Abstract
The investigation of alternating cycle-free matchings is motivated by the Jump-number problem for partially ordered sets and the problem of counting maximum cardinality matchings in hexagonal systems.
We show that the problem of deciding whether a given chordal bipartite graph has an alternating cycle-free matching of a given cardinality is NP-complete. A weaker result, for bipartite graphs only, has been known for some time. Also, the alternating cycle-free matching problem remains NP-complete for strongly chordal split graphs of diameter 2.
In contrast, we give algorithms to solve the alternating cycle-free matching problem in polynomial time for bipartite distance hereditary graphs (time O(m 2) on graphs with m edges) and distance hereditary graphs (time O(m 5)).
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Müller, H. Alternating cycle-free matchings. Order 7, 11–21 (1990). https://doi.org/10.1007/BF00383169
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DOI: https://doi.org/10.1007/BF00383169