24 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
Total -Rainbow domination numbers in graphs
Let be an integer‎, ‎and let be a graph‎. ‎A {\it‎
‎-rainbow dominating function} (or a {\it -RDF}) of is a‎
‎function from the vertex set to the family of all subsets‎
‎of such that for every with‎
‎‎, ‎the condition is fulfilled‎, ‎where is‎
‎the open neighborhood of ‎. ‎The {\it weight} of a -RDF of‎
‎ is the value ‎. ‎A -rainbow‎
‎dominating function in a graph with no isolated vertex is called‎
‎a {\em total -rainbow dominating function} if the subgraph of ‎
‎induced by the set has no isolated‎
‎vertices‎. ‎The {\em total -rainbow domination number} of ‎, ‎denoted by‎
‎‎, ‎is the minimum weight of a total -rainbow‎
‎dominating function on ‎. ‎The total -rainbow domination is the‎
‎same as the total domination‎. ‎In this paper we initiate the‎
‎study of total -rainbow domination number and we investigate its‎
‎basic properties‎. ‎In particular‎, ‎we present some sharp bounds on the‎
‎total -rainbow domination number and we determine the total‎
‎-rainbow domination number of some classes of graphs‎.