1,386 research outputs found
[1,2]-Domination in Generalized Petersen Graphs
A vertex subset of a graph is a -dominating set if each
vertex of is adjacent to either one or two vertices in . The
minimum cardinality of a -dominating set of , denoted by
, is called the -domination number of . In this
paper the -domination and the -total domination numbers of the
generalized Petersen graphs are determined
Signed double Roman domination on cubic graphs
The signed double Roman domination problem is a combinatorial optimization
problem on a graph asking to assign a label from to each
vertex feasibly, such that the total sum of assigned labels is minimized. Here
feasibility is given whenever (i) vertices labeled have at least one
neighbor with label in ; (ii) each vertex labeled has one
-labeled neighbor or at least two -labeled neighbors; and (iii) the sum
of labels over the closed neighborhood of any vertex is positive. The
cumulative weight of an optimal labeling is called signed double Roman
domination number (SDRDN). In this work, we first consider the problem on
general cubic graphs of order for which we present a sharp
lower bound for the SDRDN by means of the discharging method. Moreover, we
derive a new best upper bound. Observing that we are often able to minimize the
SDRDN over the class of cubic graphs of a fixed order, we then study in this
context generalized Petersen graphs for independent interest, for which we
propose a constraint programming guided proof. We then use these insights to
determine the SDRDNs of subcubic grid graphs, among other results
k-Tuple_Total_Domination_in_Inflated_Graphs
The inflated graph of a graph with vertices is obtained
from by replacing every vertex of degree of by a clique, which is
isomorph to the complete graph , and each edge of is
replaced by an edge in such a way that , , and
two different edges of are replaced by non-adjacent edges of . For
integer , the -tuple total domination number of is the minimum cardinality of a -tuple total dominating set
of , which is a set of vertices in such that every vertex of is
adjacent to at least vertices in it. For existing this number, must the
minimum degree of is at least . Here, we study the -tuple total
domination number in inflated graphs when . First we prove that
, and then we
characterize graphs that the -tuple total domination number number of
is or . Then we find bounds for this number in the
inflated graph , when has a cut-edge or cut-vertex , in terms
on the -tuple total domination number of the inflated graphs of the
components of or -components of , respectively. Finally, we
calculate this number in the inflated graphs that have obtained by some of the
known graphs
Rainbow domination and related problems on some classes of perfect graphs
Let and let be a graph. A function is a rainbow function if, for every vertex with
, . The rainbow domination number
is the minimum of over all rainbow
functions. We investigate the rainbow domination problem for some classes of
perfect graphs
Power domination on triangular grids
The concept of power domination emerged from the problem of monitoring
electrical systems. Given a graph G and a set S V (G), a set M of
monitored vertices is built as follows: at first, M contains only the vertices
of S and their direct neighbors, and then each time a vertex in M has exactly
one neighbor not in M, this neighbor is added to M. The power domination number
of a graph G is the minimum size of a set S such that this process ends up with
the set M containing every vertex of G. We here show that the power domination
number of a triangular grid T\_k with hexagonal-shape border of length k -- 1
is exactly $\lceil k/3 \rceil.Comment: Canadian Conference on Computational Geometry, Jul 2017, Ottawa,
Canad
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