30 research outputs found
Signed star k-domatic number of a graph
Let be a simple graph without isolated vertices with vertex set
and edge set and let be a positive integer. A function is said to be a signed star -dominating function on if
for every vertex of , where
. A set of
signed star -dominating functions on with the property that
for each , is called a signed
star -dominating family (of functions) on . The maximum number of
functions in a signed star -dominating family on is the signed
star -domatic number of , denoted by
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
Limited packings: related vertex partitions and duality issues
A -limited packing partition (LP partition) of a graph is a
partition of into -limited packing sets. We consider the LP
partitions with minimum cardinality (with emphasis on ). The minimum
cardinality is called LP partition number of and denoted by
. This problem is the dual problem of -tuple domatic
partitioning as well as a generalization of the well-studied -distance
coloring problem in graphs.
We give the exact value of for trees and bound it for
general graphs. A section of this paper is devoted to the dual of this problem,
where we give a solution to an open problem posed in . We also revisit
the total limited packing number in this paper and prove that the problem of
computing this parameter is NP-hard even for some special families of graphs.
We give some inequalities concerning this parameter and discuss the difference
between TLP number and LP number with emphasis on trees
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.