651 research outputs found
Domination in Functigraphs
Let and be disjoint copies of a graph , and let be a function. Then a \emph{functigraph}
has the vertex set and the edge set . A functigraph is a
generalization of a \emph{permutation graph} (also known as a \emph{generalized
prism}) in the sense of Chartrand and Harary. In this paper, we study
domination in functigraphs. Let denote the domination number of
. It is readily seen that . We
investigate for graphs generally, and for cycles in great detail, the functions
which achieve the upper and lower bounds, as well as the realization of the
intermediate values.Comment: 18 pages, 8 figure
Upper paired domination versus upper domination
A paired dominating set is a dominating set with the additional property
that has a perfect matching. While the maximum cardainality of a minimal
dominating set in a graph is called the upper domination number of ,
denoted by , the maximum cardinality of a minimal paired dominating
set in is called the upper paired domination number of , denoted by
. By Henning and Pradhan (2019), we know that
for any graph without isolated vertices. We
focus on the graphs satisfying the equality . We
give characterizations for two special graph classes: bipartite and unicyclic
graphs with by using the results of Ulatowski
(2015). Besides, we study the graphs with and a
restricted girth. In this context, we provide two characterizations: one for
graphs with and girth at least 6 and the other for
-free cactus graphs with . We also pose the
characterization of the general case of -free graphs with as an open question
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
On the existence of total dominating subgraphs with a prescribed additive hereditary property
AbstractRecently, Bacsó and Tuza gave a full characterization of the graphs for which every connected induced subgraph has a connected dominating subgraph satisfying an arbitrary prescribed hereditary property. Using their result, we derive a similar characterization of the graphs for which any isolate-free induced subgraph has a total dominating subgraph that satisfies a prescribed additive hereditary property. In particular, we give a characterization for the case where the total dominating subgraphs are a disjoint union of complete graphs. This yields a characterization of the graphs for which every isolate-free induced subgraph has a vertex-dominating induced matching, a so-called induced paired-dominating set
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