4,541 research outputs found
Heredity for generalized power domination
In this paper, we study the behaviour of the generalized power domination
number of a graph by small changes on the graph, namely edge and vertex
deletion and edge contraction. We prove optimal bounds for
, and for in
terms of , and give examples for which these bounds are
tight. We characterize all graphs for which for any edge . We also consider the behaviour of the
propagation radius of graphs by similar modifications.Comment: Discrete Mathematics and Theoretical Computer Science, 201
Maximum Number of Minimum Dominating and Minimum Total Dominating Sets
Given a connected graph with domination (or total domination) number
\gamma>=2, we ask for the maximum number m_\gamma and m_{\gamma,T} of
dominating and total dominating sets of size \gamma. An exact answer is
provided for \gamma=2and lower bounds are given for \gamma>=3.Comment: 6 page
Location-domination in line graphs
A set of vertices of a graph is locating if every two distinct
vertices outside have distinct neighbors in ; that is, for distinct
vertices and outside , , where
denotes the open neighborhood of . If is also a dominating set (total
dominating set), it is called a locating-dominating set (respectively,
locating-total dominating set) of . A graph is twin-free if every two
distinct vertices of have distinct open and closed neighborhoods. It is
conjectured [D. Garijo, A. Gonzalez and A. Marquez, The difference between the
metric dimension and the determining number of a graph. Applied Mathematics and
Computation 249 (2014), 487--501] and [F. Foucaud and M. A. Henning.
Locating-total dominating sets in twin-free graphs: a conjecture. The
Electronic Journal of Combinatorics 23 (2016), P3.9] respectively, that any
twin-free graph without isolated vertices has a locating-dominating set of
size at most one-half its order and a locating-total dominating set of size at
most two-thirds its order. In this paper, we prove these two conjectures for
the class of line graphs. Both bounds are tight for this class, in the sense
that there are infinitely many connected line graphs for which equality holds
in the bounds.Comment: 23 pages, 2 figure
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