Restrained domination in signed graphs

A set   r D V n G    is a restrained dominating set of   nG, where every vertex in r D )] G ( n [ V  is adjacent to a vertex in r D as well as another vertex in  )] G ( n [ V r D . The restrained domination number of lict graph   nG, denoted by   rn G  , is the minimum cardinality of a restrained dominating set of   nG. In this paper, we study its exact values for some standard graphs we obtained. Also its relation with other parameters is investigated.

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.

Discrete Mathematics, 1999

In this paper, we initiate the study of a variation of standard domination, namely restrained domination. Let G = (V; E) be a graph. A restrained dominating set is a set S ⊆ V where every vertex in V − S is adjacent to a vertex in S as well as another vertex in V − S. The restrained domination number of G, denoted by r (G), is the smallest cardinality of a restrained dominating set of G. We determine best possible upper and lower bounds for r (G); characterize those graphs achieving these bounds and ÿnd best possible upper and lower bounds for r (G) + r (G) where G is a connected graph. Finally, we give a linear algorithm for determining r (T ) for any tree and show that the decision problem for r (G) is NP-complete even for bipartite and chordal graphs.

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.

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.

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.

2011

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.

arXiv (Cornell University), 2016

A set D ⊆ V of a graph G = (V, E) is called a restrained dominating set of G if every vertex not in D is adjacent to a vertex in D and to a vertex in V \ D. The MINIMUM RESTRAINED DOMINATION problem is to find a restrained dominating set of minimum cardinality. Given a graph G, and a positive integer k, the RESTRAINED DOMINATION DECISION problem is to decide whether G has a restrained dominating set of cardinality a most k. The RESTRAINED DOMINATION DECISION problem is known to be NP-complete for chordal graphs. In this paper, we strengthen this NPcompleteness result by showing that the RESTRAINED DOMINATION DECISION problem remains NP-complete for doubly chordal graphs, a subclass of chordal graphs. We also propose a polynomial time algorithm to solve the MINIMUM RESTRAINED DOMINATION problem in block graphs, a subclass of doubly chordal graphs. The RESTRAINED DOMINATION DECISION problem is also known to be NP-complete for split graphs. We propose a polynomial time algorithm to compute a minimum restrained dominating set of threshold graphs, a subclass of split graphs. In addition, we also propose polynomial time algorithms to solve the MINIMUM RESTRAINED DOMINATION problem in cographs and chain graphs. Finally, we give a new improved upper bound on the restrained domination number, cardinality of a minimum restrained dominating set in terms of number of vertices and minimum degree of graph. We also give a randomized algorithm to find a restrained dominating set whose cardinality satisfy our upper bound with a positive probability.

Graphs and Combinatorics, 2008

The closed neighborhood N G [e] of an edge e in a graph G is the set consisting of e and of all edges having an end-vertex in common with e. Let f be a function on E(G), the edge set of G, into the set {−1, 1}. If $\sum_{x\in N[e]}f(x) \geq 1$ for each e ∈ E(G), then f is called a signed edge dominating function of G. The signed edge domination number γ s ′(G) of G is defined as $\gamma_s^\prime(G) = {\text{min}}\{\sum_{e\in E(G)}f(e)\mid f \,\text{is an SEDF of} G\}$ . Recently, Xu proved that γ s ′(G) ≥ |V(G)| − |E(G)| for all graphs G without isolated vertices. In this paper we first characterize all simple connected graphs G for which γ s ′(G) = |V(G)| − |E(G)|. This answers Problem 4.2 of [4]. Then we classify all simple connected graphs G with precisely k cycles and γ s ′(G) = 1 − k, 2 − k.

2000

We show that the efficient minus (resp., signed) domination problem is NP-complete for chordal graphs, chordal bipartite graphs, planar bipartite graphs and planar graphs of maximum degree 4 (resp., for chordal graphs). Based on the forcing property on blocks of vertices and automata theory, we provide a uniform approach to show that in a special class of interval graphs, every graph (resp., every graph with no vertex of odd degree) has an efficient minus (resp., signed) dominating function. Besides, we show that the efficient minus domination problem is equivalent to the efficient domination problem on trees.

European Journal of Pure and Applied Mathematics, 2021

Let G be a connected graph. Brigham et al. [3] defined a resolving dominating setas a set S of vertices of a connected graph G that is both resolving and dominating. A set S ⊆ V (G) is a resolving restrained dominating set of G if S is a resolving dominating set of G and S = V (G) or hV (G) \ Si has no isolated vertex. In this paper, we characterize the resolving restrained dominating sets in the join, corona and lexicographic product of graphs and determine the resolving restrained domination number of these graphs.

RAIRO - Operations Research

A restrained {2}-dominating function (R{2}DF) on a graph G = (V, E) is a function f : V → {0, 1, 2} such that : (i) f(N[v]) ≥ 2 for all v ∈ V, where N[v] is the set containing v and all vertices adjacent to v; (ii) the subgraph induced by the vertices assigned 0 under f has no isolated vertices. The weight of an R{2}DF is the sum of its function values over all vertices, and the restrained {2}-domination number γr{2}(G) is the minimum weight of an R{2}DF on G. In this paper, we initiate the study of the restrained {2}-domination number. We first prove that the problem of computing this parameter is NP-complete, even when restricted to bipartite graphs. Then we give various bounds on this parameter. In particular, we establish upper and lower bounds on the restrained {2}-domination number of a tree T in terms of the order, the numbers of leaves and support vertices.

Information Processing Letters, 2009

In this paper we present unified methods to solve the minus and signed total domination problems for chordal bipartite graphs and trees in O (n 2) and O (n + m) time, respectively. We also prove that the decision problem for the signed total domination problem on doubly chordal graphs is NP-complete. Note that bipartite permutation graphs, biconvex bipartite graphs, and convex bipartite graphs are subclasses of chordal bipartite graphs.

2016

Let G be a connected nontrivial graph. A nonempty subset S of V (G) is a 1movable restrained dominating set of G if S is a restrained dominating set of G and for every v ∈ S, S \ {v} is a restrained dominating set of G or there exists u ∈ (V (G) \ S) ∩ NG(v) such that (S \ {v}) ∪ {u} is a restrained dominating set of G. The 1-movable restrained domination number of a graph G, denoted by γ 1 mr(G), is the cardinality of the smallest 1-movable restrained dominating set of G. A 1-movable restrained dominating set of G with cardinality equal to γ 1 mr(G) is called γ 1 mr -set of G. This paper presents some properties of 1-movable restrained dominating set and investigates the 1-movable restrained dominating sets in the join of two graphs. Moreover, the bounds or exact values of the 1-movable restrained domination number are determined. AMS subject classification: 05C69.

2016

A function f : V → {−1, 0, 1} is an affirmative dominating function of graph G satisfying the conditions that for every vertex u such that f(u) = 0 is adjacent to at least one vertex v for which f(v) = 1 and P u∈N(v) f(u) ≤ 1 for every v ∈ V . The affirmative domination number γa(G) =max{w(f) : f is affirmative dominating function}. In this paper, we initiate the study of affirmative and strongly affirmative dominating functions. Here, we obtain some properties of these new parameters and also determine exact values of some special classes of graph.