An upper bound for domination number of 5-regular graphs

Let G = (V, E) be a simple graph. A subset S ⊆ V is a dominating set of G, if for any vertex u ∈ V-S, there exists a vertex v ∈ S such that uv ∈ E. The domination number, denoted by γ(G), is the minimum cardinality of a dominating set. In this paper we will prove that if G is a 5-regular graph, then γ(G) ⩽ 5/14n.     

On Trees with Double Domination Number Equal to the 2-Outer-Independent Domination Number Plus One

A vertex of a graph is said to dominate itself and all of its neighbors.A double dominating set of a graph G is a set D of vertices of G,such that every vertex of G is dominated by at least two vertices of D.The double domination number of a graph G is the minimum cardinality of a double dominating set of G.For a graph G =(V,E),a subset D V(G) is a 2-dominating set if every vertex of V(G) \ D has at least two neighbors in D,while it is a 2-outer-independent dominating set of G if additionally the set V(G)\D is independent.The 2-outer-independent domination number of G is the minimum cardinality of a 2-outer-independent dominating set of G.This paper characterizes all trees with the double domination number equal to the 2-outer-independent domination number plus one.     

Domination number in graphs with minimum degree two

A set D of vertices of a graph G = （V, E） is called a dominating set if every vertex of V not in D is adjacent to a vertex of D. In 1996, Reed proved that every graph of order n with minimum degree at least 3 has a dominating set of cardinality at most 3n/8. In this paper we generalize Reed＇s result. We show that every graph G of order n with minimum degree at least 2 has a dominating set of cardinality at most （3n ＋IV21）/8, where V2 denotes the set of vertices of degree 2 in G. As an application of the above result, we show that for k ≥ 1, the k-restricted domination number rk （G, γ） ≤ （3n＋5k）/8 for all graphs of order n with minimum degree at least 3.     

Bounds on the Distance Two-Domination Number of a Graph

 For a graph G = (V, E), a subset D⊆V(G) is said to be distance two-dominating set in G if for each vertex u∈V−D, there exists a vertex v∈D such that d(u,v)≤2. The minimum cardinality of a distance two-dominating set in G is called a distance two-domination number and is denoted by γ₂(G). In this note we obtain various upper bounds for γ₂(G) and characterize the classes of graphs attaining these bounds. Received: May 31, 1999 Final version received: July 13, 2000     

On the Structure of Graphs with Bounded Asteroidal Number

 A set A⊆V of the vertices of a graph G=(V,E) is an asteroidal set if for each vertex a∈A, the set A\{a} is contained in one component of G−N[a]. The maximum cardinality of an asteroidal set of G, denoted by an (G), is said to be the asteroidal number of G. We investigate structural properties of graphs of bounded asteroidal number. For every k≥1, an (G)≤k if and only if an (H)≤k for every minimal triangulation H of G. A dominating target is a set D of vertices such that D∪S is a dominating set of G for every set S such that G[D∪S] is connected. We show that every graph G has a dominating target with at most an (G) vertices. Finally, a connected graph G has a spanning tree T such that d _T (x,y)−d _G (x,y)≤3·|D|−1 for every pair x,y of vertices and every dominating target D of G. Received: July 3, 1998 Final version received: August 10, 1999     

Total outer-connected domination numbers of trees

Let G=(V,E) be a graph without an isolated vertex. A set DV(G) is a total dominating set if D is dominating, and the induced subgraph G[D] does not contain an isolated vertex. The total domination number of G is the minimum cardinality of a total dominating set of G. A set DV(G) is a total outer-connected dominating set if D is total dominating, and the induced subgraph G[V(G)−D] is a connected graph. The total outer-connected domination number of G is the minimum cardinality of a total outer-connected dominating set of G. We characterize trees with equal total domination and total outer-connected domination numbers. We give a lower bound for the total outer-connected domination number of trees and we characterize the extremal trees.     

Trees with Equal Domination and Restrained Domination Numbers

Let G = (V,E) be a graph and let S V. The set S is a packing in G if the vertices of S are pairwise at distance at least three apart in G. The set S is a dominating set (DS) if every vertex in V − S is adjacent to a vertex in S. Further, if every vertex in V − S is also adjacent to a vertex in V − S, then S is a restrained dominating set (RDS). The domination number of G, denoted by γ(G), is the minimum cardinality of a DS of G, while the restrained domination number of G, denoted by γ_r(G), is the minimum cardinality of a RDS of G. The graph G is γ-excellent if every vertex of G belongs to some minimum DS of G. A constructive characterization of trees with equal domination and restrained domination numbers is presented. As a consequence of this characterization we show that the following statements are equivalent: (i) T is a tree with γ(T)=γ_r(T); (ii) T is a γ-excellent tree and T ≠ K₂; and (iii) T is a tree that has a unique maximum packing and this set is a dominating set of T. We show that if T is a tree of order n with ℓ leaves, then γ_r(T) ≤ (n + ℓ + 1)/2, and we characterize those trees achieving equality.     

A linear algorithm for the domination number of a series-parallel graph

A vertex u in an undirected graph G = (V, E) is said to dominate all its adjacent vertices and itself. A subset D of V is a dominating set in G if every vertex in G is dominated by a vertex in D, and is a minimum dominating set in G if no other dominating set in G has fewer vertices than D. The domination number of G is the cardinality of a minimum dominating set in G.The problem of determining, for a given positive integer k and an undirected graph G, whether G has a dominating set D in G satisfying ¦D¦ ≤ k, is a well-known NP-complete problem. Cockayne have presented a linear time algorithm for finding a minimum dominating set in a tree. In this paper, we will present a linear time algorithm for finding a minimum dominating set in a series-parallel graph.     

Bounds on the Total Restrained Domination Number of a Graph

Let G = (V, E) be a graph. A set S í V{S \subseteq V} is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of V − S is adjacent to a vertex in V − S. The total restrained domination number of G, denoted by γ _tr (G), is the smallest cardinality of a total restrained dominating set of G. We show that if δ ≥ 3, then γ _tr (G) ≤ n − δ − 2 provided G is not one of several forbidden graphs. Furthermore, we show that if G is r − regular, where 4 ≤ r ≤ n − 3, then γ _tr (G) ≤ n − diam(G) − r + 1.     

On Domination Number of 4-Regular Graphs

Let G be a simple graph. A subset S V is a dominating set of G, if for any vertex v V – S there exists a vertex u S such that uv E(G). The domination number, denoted by (G), is the minimum cardinality of a dominating set. In this paper we prove that if G is a 4-regular graph with order n, then (G) ⁴/₁₁ n     

Minimum 2-tuple dominating set of permutation graphs

For a fixed positive integer k, a k-tuple dominating set of a graph G=(V,E) is a subset D?V such that every vertex in V is dominated by at least k vertex in D. The k-tuple domination number γ _×k (G) is the minimum size of a k-tuple dominating set of G. The special case when k=1 is the usual domination. The case when k=2 was called double domination or 2-tuple domination. A 2-tuple dominating set D ₂ is said to be minimal if there does not exist any D′?D ₂ such that D′ is a 2-tuple dominating set of G. A 2-tuple dominating set D ₂, denoted by γ _×2(G), is said to be minimum, if it is minimal as well as it gives 2-tuple domination number. In this paper, we present an efficient algorithm to find a minimum 2-tuple dominating set on permutation graphs with n vertices which runs in O(n ²) time.     

Minus total domination in graphs

A three-valued function f: V → {−1, 0, 1} defined on the vertices of a graph G= (V, E) is a minus total dominating function (MTDF) if the sum of its function values over any open neighborhood is at least one. That is, for every υ ∈ V, f(N(υ)) ⩾ 1, where N(υ) consists of every vertex adjacent to υ. The weight of an MTDF is f(V) = Σf(υ), over all vertices υ ∈ V. The minus total domination number of a graph G, denoted γ _t ⁻(G), equals the minimum weight of an MTDF of G. In this paper, we discuss some properties of minus total domination on a graph G and obtain a few lower bounds for γ _t ⁻(G).     

Total domination in graphs with minimum degree three

A set S of vertices of a graph G is a total dominating set, if every vertex of V(G) is adjacent to some vertex in S. The total domination number of G, denoted by γ_t(G), is the minimum cardinality of a total dominating set of G. We prove that, if G is a graph of order n with minimum degree at least 3, then γ_t(G) ≤ 7n/13. © 2000 John Wiley & Sons, Inc. J Graph Theory 34:9–19, 2000     

An Upper Bound for the Total Domination Subdivision Number of a Graph

A set S of vertices of a graph G = (V, E) without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number γ _t (G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sd_gt(G){{\rm sd}_{\gamma_t}(G)} is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. In this paper, we prove that sd_gt(G) ￡ 2g_t(G)-1{{\rm sd}_{\gamma_t}(G)\leq 2\gamma_t(G)-1} for every simple connected graph G of order n ≥ 3.     

A Greedy Partition Lemma for directed domination

A directed dominating set in a directed graph D is a set S of vertices of V such that every vertex u∈V(D)?S has an adjacent vertex v in S with v directed to u. The directed domination number of D, denoted by γ(D), is the minimum cardinality of a directed dominating set in D. The directed domination number of a graph G, denoted Γ_d(G), is the maximum directed domination number γ(D) over all orientations D of G. The directed domination number of a complete graph was first studied by Erd?s [P. Erd?s On a problem in graph theory, Math. Gaz. 47 (1963) 220–222], albeit in a disguised form. In this paper we prove a Greedy Partition Lemma for directed domination in oriented graphs. Applying this lemma, we obtain bounds on the directed domination number. In particular, if α denotes the independence number of a graph G, we show that α≤Γ_d(G)≤α(1+2ln(n/α)).     

Characterizations of trees with equal domination parameters

Let G = (V, E) be a graph. A set S ⊆ V is a restrained dominating set, if every vertex not in S is adjacent to a vertex in S and to a vertex in V − S. The restrained domination number of G, denoted by γ_r(G), is the minimum cardinality of a restrained dominating set of G. A set S ⊆ V is a weak dominating set of G if, for every u in V − S, there exists a v ∈ S such that uv ∈ E and deg u ≥ deg v. The weak domination number of G, denoted by γ_w(G), is the minimum cardinality of a weak dominating set of G. In this article, we provide a constructive characterization of those trees with equal independent domination and restrained domination numbers. A constructive characterization of those trees with equal independent domination and weak domination numbers is also obtained. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 142–153, 2000     

A Generalization of the Gallai–Roy Theorem

 A well-known and essential result due to Roy ([4], 1967) and independently to Gallai ([3], 1968) is that if D is a digraph with chromatic number χ(D), then D contains a directed path of at least χ(D) vertices. We generalize this result by showing that if ψ(D) is the minimum value of the number of the vertices in a longest directed path starting from a vertex that is connected to every vertex of D, then χ(D) ≤ψ(D). For graphs, we give a positive answer to the following question of Fajtlowicz: if G is a graph with chromatic number χ(G), then for any proper coloring of G of χ(G) colors and for any vertex v∈V(G), there is a path P starting at v which represents all χ(G) colors. Received: May 20, 1999 Final version received: December 24, 1999     

Bounds on Laplacian eigenvalues related to total and signed domination of graphs

A total dominating set in a graph G is a subset X of V (G) such that each vertex of V (G) is adjacent to at least one vertex of X. The total domination number of G is the minimum cardinality of a total dominating set. A function f: V (G) → {−1, 1} is a signed dominating function (SDF) if the sum of its function values over any closed neighborhood is at least one. The weight of an SDF is the sum of its function values over all vertices. The signed domination number of G is the minimum weight of an SDF on G. In this paper we present several upper bounds on the algebraic connectivity of a connected graph in terms of the total domination and signed domination numbers of the graph. Also, we give lower bounds on the Laplacian spectral radius of a connected graph in terms of the signed domination number of the graph.     

Dominating Sets of Some Graphs Associated to Commutative Rings

Let G = (V, E) be a graph. A set S ? V is a dominating set of G if every vertex in V is either in S or is adjacent to a vertex in S. The domination number γ(G) of G is the minimum cardinality among the dominating sets of G. The main object of this article is to study and characterize the dominating sets of the zero-divisor graph Γ(R) and ideal-based zero-divisor graph Γ _I (R) of a commutative ring R.     

Locating and total dominating sets in trees

A set S of vertices in a graph G=(V,E) is a total dominating set of G if every vertex of V is adjacent to a vertex in S. We consider total dominating sets of minimum cardinality which have the additional property that distinct vertices of V are totally dominated by distinct subsets of the total dominating set.