11 research outputs found
Discrepancy and Signed Domination in Graphs and Hypergraphs
For a graph G, a signed domination function of G is a two-colouring of the
vertices of G with colours +1 and -1 such that the closed neighbourhood of
every vertex contains more +1's than -1's. This concept is closely related to
combinatorial discrepancy theory as shown by Fueredi and Mubayi [J. Combin.
Theory, Ser. B 76 (1999) 223-239]. The signed domination number of G is the
minimum of the sum of colours for all vertices, taken over all signed
domination functions of G. In this paper, we present new upper and lower bounds
for the signed domination number. These new bounds improve a number of known
results.Comment: 12 page
The Signed Domination Number of Cartesian Products of Directed Cycles
Let D be a finite simple directed graph with vertex set V(D) and arc set A(D). A function is called a signed dominating function (SDF) iffor each vertex v. The weight w(f ) of f is defined by. The signed domination number of a digraph D is gs(D) = min{w(f ) : f is an SDF of D}. Let Cmn denotes the Cartesian product of directed cycles of length m and n. In this paper, we determine the exact value of signed domination number of some classes of Cartesian product of directed cycles. In particular, we prove that: (a) gs(C3n) = n if n 0(mod 3), otherwise gs(C3n) = n + 2. (b) gs(C4n) = 2n. (c) gs(C5n) = 2n if n 0(mod 10), gs(C5n) = 2n + 1 if n 3, 5, 7(mod 10), gs(C5n) = 2n + 2 if n 2, 4, 6, 8(mod 10), gs(C5n) = 2n + 3 if n 1,9(mod 10). (d) gs(C6n) = 2n if n 0(mod 3), otherwise gs(C6n) = 2n + 4. (e) gs(C7n) = 3n
On the total signed domination number of the Cartesian product of paths
Let be a finite connected simple graph with a vertex set and an edge set . A total signed dominating function of is a function , such that for all . The total signed domination number of is the minimum weight of a total signed dominating function on . In this paper, we prove lower and upper bounds on the total signed domination number of the Cartesian product of two paths,
Bounds on the signed distance--domination number of graphs
Abstract Let , be a graph with vertex set of order and edge set . A -dominating set of is a subset such that each vertex in \ has at least neighbors in . If is a vertex of a graph , the open -neighborhood of , denoted by , is the set , . is the closed -neighborhood of . A function 1, 1 is a signed distance--dominating function of , if for every vertex , ∑ 1. The signed distance--domination number, denoted by , , is the minimum weight of a signed distance--dominating function of . In this paper, we give lower and upper bounds on , of graphs. Also, we determine the signed distance--domination number of graph , (the graph obtained from the disjoint union by adding the edges , ) when 2
A Survey on Alliances and Related Parameters in Graphs
In this paper, we show that several graph parameters are known in different areas under completely different names.More specifically, our observations connect signed domination, monopolies, -domination, -independence,positive influence domination,and a parameter associated to fast information propagationin networks to parameters related to various notions of global -alliances in graphs.We also propose a new framework, called (global) -alliances, not only in order to characterizevarious known variants of alliance and domination parameters, but also to suggest a unifying framework for the study of alliances and domination.Finally, we also give a survey on the mentioned graph parameters, indicating how results transfer due to our observations