230 research outputs found
The Signed Roman Domatic Number of a Digraph
Let be a finite and simple digraph with vertex set .A {\em signed Roman dominating function} on the digraph isa function such that for every , where consists of andall inner neighbors of , and every vertex for which has an innerneighbor for which . A set of distinct signedRoman dominating functions on with the property that for each, is called a {\em signed Roman dominating family} (of functions) on . The maximumnumber of functions in a signed Roman dominating family on is the {\em signed Roman domaticnumber} of , denoted by . In this paper we initiate the study of signed Romandomatic number in digraphs and we present some sharp bounds for . In addition, wedetermine the signed Roman domatic number of some digraphs. Some of our results are extensionsof well-known properties of the signed Roman domatic number of graphs
Signed total double Roman dominatıon numbers in digraphs
Let D = (V, A) be a finite simple digraph. A signed total double Roman dominating function (STDRD-function) on the digraph D is a function f : V (D) → {−1, 1, 2, 3} satisfying the following conditions: (i) P x∈N−(v) f(x) ≥ 1 for each v ∈ V (D), where N−(v) consist of all in-neighbors of v, and (ii) if f(v) = −1, then the vertex v must have at least two in-neighbors assigned 2 under f or one in-neighbor assigned 3 under f, while if f(v) = 1, then the vertex v must have at least one in-neighbor assigned 2 or 3 under f. The weight of a STDRD-function f is the value P x∈V (D) f(x). The signed total double Roman domination number (STDRD-number) γtsdR(D) of a digraph D is the minimum weight of a STDRD-function on D. In this paper we study the STDRD-number of digraphs, and we present lower and upper bounds for γtsdR(D) in terms of the order, maximum degree and chromatic number of a digraph. In addition, we determine the STDRD-number of some classes of digraphs.Publisher's Versio
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
- …