Graphs of separability at most two: structural characterizations and their consequences

Authors:
Ferdinando Cicalese;Martin Milanič
Affiliations:
Dipartimento di Informatica ed Applicazioni, University of Salerno, Fisciano, Italy;FAMNIT and PINT, University of Primorska, Koper, Slovenia
Venue:
IWOCA'10 Proceedings of the 21st international conference on Combinatorial algorithms
Year:
2010

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Abstract

Graphs of separability at most k are defined as graphs in which every two non-adjacent vertices are separated by a set of at most k other vertices. For k ∈ {0,1}, the only connected graphs of separability at most k are complete graphs and block graphs, respectively. For k ≥ 3, graphs of separability at most k form a rich class of graphs containing all graphs of maximum degree k. Graphs of separability at most 2 generalize complete graphs, cycles and trees. We prove several characterizations of graphs of separability at most 2 and examine some of their consequences.