18,845 research outputs found
A note on the independent roman domination in unicyclic graphs
A Roman dominating function (RDF) on a graph is a function satisfying the condition that every vertex for which is adjacent to at least one vertex for which . The weight of an RDF is the value . An RDF in a graph is independent if no two vertices assigned positive values are adjacent. The Roman domination number (respectively, the independent Roman domination number ) is the minimum weight of an RDF (respectively, independent RDF) on . We say that strongly equals , denoted by , if every RDF on of minimum weight is independent. In this note we characterize all unicyclic graphs with
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
Trees with Unique Italian Dominating Functions of Minimum Weight
An Italian dominating function, abbreviated IDF, of is a function satisfying the condition that for every vertex with , we have . That is, either is adjacent to at least one vertex with , or to at least two vertices and with . The Italian domination number, denoted (G), is the minimum weight of an IDF in . In this thesis, we use operations that join two trees with a single edge in order to build trees with unique -functions
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