Exact leaf powers

Authors:
Andreas Brandstädt;Van Bang Le;Dieter Rautenbach
Affiliations:
Institut für Informatik, Universität Rostock, D-18051 Rostock, Germany;Institut für Informatik, Universität Rostock, D-18051 Rostock, Germany;Institut für Mathematik, TU Ilmenau, Postfach 100565, D-98684 Ilmenau, Germany
Venue:
Theoretical Computer Science
Year:
2010

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Abstract

We define and study the new notion of exact k-leaf powers where a graph G=(V"G,E"G) is an exact k-leaf power if and only if there exists a tree T=(V"T,E"T) - an exact k-leaf root of G - whose set of leaves equals V"G such that uv@?E"G holds for u,v@?V"G if and only if the distance of u and v in T is exactly k. This new notion is closely related to but different from leaf powers and neighbourhood subtree tolerance graphs. We prove characterizations of exact 3- and 4-leaf powers which imply that such graphs can be recognized in linear time and that also the corresponding exact leaf roots can be found in linear time. Furthermore, we characterize all exact 5-leaf roots of chordless cycles and derive several properties of exact 5-leaf powers.