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Disjoint quasi‐kernels in digraphs
Author(s) -
Heard Scott,
Huang Jing
Publication year - 2008
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.20310
Subject(s) - digraph , combinatorics , disjoint sets , mathematics , vertex (graph theory) , transitive relation , multipartite , discrete mathematics , kernel (algebra) , graph , physics , quantum mechanics , quantum entanglement , quantum
A quasi‐kernel in a digraph is an independent set of vertices such that any vertex in the digraph can reach some vertex in the set via a directed path of length at most two. Chvátal and Lovász proved that every digraph has a quasi‐kernel. Recently, Gutin et al. raised the question of which digraphs have a pair of disjoint quasi‐kernels. Clearly, a digraph has a pair of disjoint quasi‐kernels cannot contain sinks , that is, vertices of outdegree zero, as each such vertex is necessarily included in a quasi‐kernel. However, there exist digraphs which contain neither sinks nor a pair of disjoint quasi‐kernels. Thus, containing no sinks is not sufficient in general for a digraph to have a pair of disjoint quasi‐kernels. In contrast, we prove that, for several classes of digraphs, the condition of containing no sinks guarantees the existence of a pair of disjoint quasi‐kernels. The classes contain semicomplete multipartite, quasi‐transitive, and locally semicomplete digraphs. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:251‐260, 2008