2,474 research outputs found
Disjoint Dominating Sets with a Perfect Matching
In this paper, we consider dominating sets and such that and
are disjoint and there exists a perfect matching between them. Let
denote the cardinality of smallest such sets in
(provided they exist, otherwise ). This
concept was introduced in [Klostermeyer et al., Theory and Application of
Graphs, 2017] in the context of studying a certain graph protection problem. We
characterize the trees for which equals a certain
graph protection parameter and for which ,
where is the independence number of . We also further study this
parameter in graph products, e.g., by giving bounds for grid graphs, and in
graphs of small independence number
Partitioning the vertex set of to make an efficient open domination graph
A graph is an efficient open domination graph if there exists a subset of
vertices whose open neighborhoods partition its vertex set. We characterize
those graphs for which the Cartesian product is an efficient
open domination graph when is a complete graph of order at least 3 or a
complete bipartite graph. The characterization is based on the existence of a
certain type of weak partition of . For the class of trees when is
complete of order at least 3, the characterization is constructive. In
addition, a special type of efficient open domination graph is characterized
among Cartesian products when is a 5-cycle or a 4-cycle.Comment: 16 pages, 2 figure
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