We study the computational complexity of partitioning the vertices of a graph into generalized dominating sets. Generalized dominating sets are param-eterized by two sets of nonnegative integers and which constrain the neighborhood N (v) of vertices. A set S of vertices of a graph is said to be a (;)-set if 8v 2 S : jN(v)\Sj 2 and 8v 6 2 S : jN(v)\Sj 2 .

Let D be a finite simple directed graph with vertex set V(D) and arc set A(D). A function ( ) { } − : 1,1 f V D → is called a signed dominating function (SDF) if [ ] ( ) 1 D f N v − ≥ for each vertex v V ∈ . The weight ( ) f ω of f is defined by ( ) ∑ v V f v ∈ . The signed domination number of a digraph D is ( ) ( ) { } γ ω = min is an SDF of s D f f D . Let Cm × Cn denotes the cartesian produ...

We study the computational complexity of partitioning the vertices of a graph into generalized dominating sets. Generalized dominating sets are parameterized by two sets of nonnegative integers and which constrain the neighborhood N (v) of vertices. A set S of vertices of a graph is said to be a (;)-set if 8v 2 S : jN(v) \ Sj 2 and 8v 6 2 S : jN(v) \ Sj 2 .

This paper is devoted to the study of quadruple Roman domination in trees, and it a contribution Special Issue “Theoretical computer science discrete mathematics” Symmetry. For any positive integer k, [k]-Roman dominating function ([k]-RDF) simple graph G from vertex set V {0,1,2,…,k+1} if for u?V with f(u)<k, ?x?N(u)?{u}f(x)?|{x?N(u):f(x)?1}|+k, where N(u) open neighborhood u. The weight [k...

Let G = (V,E) be a graph and let f be a function f : E → {0, 1, 2}. An edge x with f(x) = 0 is said to be undefended with respect to f if it is not incident to an edge with positive weight. The function f is a weak edge Roman dominating function (WERDF) if each edge x with f(x) = 0 is incident to an edge y with f(y) > 0 such that the function f ′ : E → {0, 1, 2}, defined by f ′(x) = 1, f ′(y) =...

This paper is motivated by the concept of the signed k-domination problem and dedicated to the complexity of the problem on graphs. For any fixed nonnegative integer k, we show that the signed k-domination problem is NP-complete for doubly chordal graphs. For strongly chordal graphs and distance-hereditary graphs, we show that the signed k-domination problem can be solved in polynomial time. We...

Dominating sets in their many variations model a wealth of optimization problems like facility location or distributed le sharing. For instance, when a request can occur at any node in a graph and requires a server at that node, a minimumdominating set represents a minimum set of servers that serve an arbitrary single request by moving a server along at most one edge. This paper studies dominat...