Signed total domination in graphs

Discrete Applied Mathematics, 2009

a b s t r a c t Let G = (V, E) be a graph. A function f : V → {−1, +1} defined on the vertices of G is a signed total dominating function if the sum of its function values over any open neighborhood is at least one. A signed total dominating function f is minimal if there does not exist a signed total dominating function

Publicationes Mathematicae Debrecen, 2011

A function f : V (G) → {−1, 1} defined on the vertices of a graph G is a signed total dominating function (STDF) if the sum of its function values over any open neighborhood is at least one. A STDF f of G is called a global signed total dominating function (GSTDF) if f is also a STDF of the complement G of G. The global signed total domination number γgst(G) of G is defined as γgst(G) = min{ v∈V (G) f (v) | f is a GSTDF of G}. In this paper first we find lower and upper bounds for the global signed total domination number of a graph. Then we prove that if T is a tree of order n ≥ 4 with ∆(T) ≤ n − 2, then γ gst (T) ≤ γ st (T) + 4. We characterize all the trees which satisfy the equality. We also characterize all trees T of order n ≥ 4, ∆(T) ≤ n − 2 and γ gst (T) = γ st (T) + 2.

Opuscula Mathematica, 2016

Let G = (V, E) be a simple graph. A function f : V → {−1, 1} is called an inverse signed total dominating function if the sum of its function values over any open neighborhood is at most zero. The inverse signed total domination number of G, denoted by γ 0 st (G), equals to the maximum weight of an inverse signed total dominating function of G. In this paper, we establish upper bounds on the inverse signed total domination number of graphs in terms of their order, size and maximum and minimum degrees.

Electronic Journal of Graph Theory and Applications, 2016

A nonnegative signed dominating function (NNSDF) of a graph G is a function f from the vertex set V (G) to the set {−1, 1} such that u∈N [v] f (u) ≥ 0 for every vertex v ∈ V (G). The nonnegative signed domination number of G, denoted by γ N N s (G), is the minimum weight of a nonnegative signed dominating function on G. In this paper, we establish some sharp lower bounds on the nonnegative signed domination number of graphs in terms of their order, size and maximum and minimum degree.

Electronic Notes in Discrete Mathematics, 2002

Let G = (V , E) be a simple graph on vertex set V and define a function f : V → {−1, 1}. The function f is a signed dominating function if for every vertex x ∈ V , the closed neighborhood of x contains more vertices with function value 1 than with −1. The signed domination number of G, γ s (G), is the minimum weight of a signed dominating function on G. We give a sharp lower bound on the signed domination number of a general graph with a given minimum and maximum degree, generalizing a number of previously known results. Using similar techniques we give upper and lower bounds for the signed domination number of some simple graph products: the grid P j × P k , C j × P k and C j × C k . For fixed width, these bounds differ by only a constant.

Czechoslovak Mathematical Journal, 2008

The open neighborhood N G (e) of an edge e in a graph G is the set consisting of all edges having a common end-vertex with e.

1995

A two-valued function f defined on the vertices of a graph G (V, E), I : V-+ {-I, I}, is a signed dominating function if the sum of its function values over any closed neighborhood is at least one. That is, for every v E V, f(N[v]) 2: 1, where N(v] consists of v and every vertex adjacent to v. The of a signed dominating function is ICV) = L f(v), over all vertices v E V. The signed domination number of graph G, denoted /s(G), equals the minimum weight of a signed dominating function of G. The upper signed domination number of a graph G, denoted r.(G), equals the maximum weight of a minimal signed dominating function of G. In this paper we present a variety of algorithmic results on the complexity of signed and upper signed domination in graphs.

2004

A two-valued function f defined on the vertices of a graph G = (V, E), f : V → {-1, 1}, is a signed total dominating function if the sum of its function values over any open neighborhood is at least one. That is, for every v ∈ V, f (N (v)) ≥ 1, where N (v) consists of every vertex adjacent to v. The weight of a total signed dominating function is f (V ) = f (v), over all vertices v ∈ V . The total signed domination number of a graph G, denoted γ s t (G), equals the minimum weight of a total signed dominating function of G. If, instead of the range {-1, 1}, we allow the range {-1, 0, 1}, then we get the concept of a total minus dominating function. Its associated parameter, called the total minus domination number of a graph G, is denoted γ - t (G). In this paper, we show that the decision problem corresponding to the computation of the total minus domination number of a graph is NP-complete, even when restricted to bipartite graphs or chordal graphs. For a fixed k, we show that the decision problem corresponding to determining whether a graph has a total minus dominating function of weight at most k may be NPcomplete, even when restricted to bipartite or chordal graphs. Linear time algorithms for computing γ - t (T ) and γ s t (T ) for an arbitrary tree T are also presented.

2011

v∈V f(v). The complementary signed domination number of G is defined as γcs(G) = min {w(f) : f is a minimal complementary signed dominating function of G}. In this paper, we determine the value of complementary signed domination number for some special class of graphs. We also determine bounds for this parameter and exhibit the sharpness of the bounds. We also characterize graphs attaining the bounds in some special classes.

Discrete Mathematics, 2004

Let G = (V, E) be a simple graph on vertex set V and define a function f : V → {−1, 1}. The function f is a signed dominating function if for every vertex x ∈ V , the closed neighborhood of x contains more vertices with function value 1 than with −1. The signed domination number of G, γ s (G), is the minimum weight of a signed dominating function on G.