Total restrained domination in graphs with minimum degree two

In this paper, we continue the study of total restrained domination in graphs, a concept introduced by Telle and Proskurowksi (Algorithms for vertex partitioning problems on partial k-trees, SIAM J. Discrete Math. 10 (1997) 529-550) as a vertex partitioning problem. A set S of vertices in a graph G=(V,E) is a total restrained dominating set of G 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 minimum cardinality of a total restrained dominating set of G is the total restrained domination number of G, denoted by γ_tr(G). Let G be a connected graph of order n with minimum degree at least 2 and with maximum degree Δ where Δ?n-2. We prove that if n?4, then and this bound is sharp. If we restrict G to a bipartite graph with Δ?3, then we improve this bound by showing that and that this bound is sharp.     

Total domination in claw-free graphs with minimum degree 2

A set S of vertices in a graph G is a total dominating set (TDS) of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a TDS of G is the total domination number of G, denoted by γ_t(G). A graph is claw-free if it does not contain K_1,3 as an induced subgraph. It is known [M.A. Henning, Graphs with large total domination number, J. Graph Theory 35(1) (2000) 21-45] that if G is a connected graph of order n with minimum degree at least two and G∉{C₃,C₅, C₆, C₁₀}, then γ_t(G)?4n/7. In this paper, we show that this upper bound can be improved if G is restricted to be a claw-free graph. We show that every connected claw-free graph G of order n and minimum degree at least two satisfies γ_t(G)?(n+2)/2 and we characterize those graphs for which γ_t(G)=⌊(n+2)/2⌋.     

Irredundance, secure domination and maximum degree in trees

It is shown that the lower irredundance number and secure domination number of an n vertex tree T with maximum degree Δ?3, are bounded below by 2(n+1)/(2Δ+3)(T≠K_1,Δ) and (Δn+Δ-1)/(3Δ-1), respectively. The bounds are sharp and extremal trees are exhibited.     

A note on the girth of digraphs

Behzad, Chartrand and Wall conjectured that the girth of a diregular graph of ordern and outdegreer is not greater than [n /r]. This conjecture has been proved forr=2 by Behzad and forr=3 by Bermond. We prove that a digraph of ordern and halfdegree ≧4 has girth not exceeding [n / 4]. We also obtain short proofs of the above results. Our method is an application of the theory of connectivity of digraphs.     

Graphs with prescribed degree sets and girth

For a finite nonempty set of integers, each of which is at least two, and an integern 3, the numberf(;n) is defined as the least order of a graph having degree set and girthn. The numberf(;n) is evaluated for several sets and integersn. In particular, it is shown thatf({3, 4}; 5) = 13 andf({3, 4}; 6) = 18.Research of the third author was partially supported by a Faculty Research Fellowship from Western Michigan University.     

Total domination and total domination subdivision number of a graph and its complement

A set S of vertices of a graph G=(V,E) with no isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination numberγ_t(G) is the minimum cardinality of a total dominating set of G. The total domination subdivision numbersd_γt(G) is the minimum number of edges that must be subdivided in order to increase the total domination number. We consider graphs of order n?4, minimum degree δ and maximum degree Δ. We prove that if each component of G and has order at least 3 and , then and if each component of G and has order at least 2 and at least one component of G and has order at least 3, then . We also give a result on stronger than a conjecture by Harary and Haynes.     

Bounds on cost effective domination numbers

A vertex υ in a set S is said to be cost effective if it is adjacent to at least as many vertices in V\S as it is in S and is very cost effective if it is adjacent to more vertices in V\S than to vertices in S. A dominating set S is (very) cost effective if every vertex in S is (very) cost effective. The minimum cardinality of a (very) cost effective dominating set of G is the (very) cost effective domination number of G. Our main results include a quadratic upper bound on the very cost effective domination number of a graph in terms of its domination number. The proof of this result gives a linear upper bound for hereditarily sparse graphs which include trees. We show that no such linear bound exists for graphs in general, even when restricted to bipartite graphs. Further, we characterize the extremal trees attaining the bound. Noting that the very cost effective domination number is bounded below by the domination number, we show that every value of the very cost effective domination number between these lower and upper bounds for trees is realizable. Similar results are given for the cost effective domination number.     

Stratification and domination in graphs II

A graph G is 2-stratified if its vertex set is partitioned into two classes (each of which is a stratum or a color class.) We color the vertices in one color class red and the other color class blue. Let F be a 2-stratified graph with one fixed blue vertex v specified. We say that F is rooted at v. The F-domination number of a graph G is the minimum number of red vertices of G in a red-blue coloring of the vertices of G such that every blue vertex v of G belongs to a copy of F rooted at v. In this paper we investigate the F-domination number when (i) F is a 2-stratified path P₃ on three vertices rooted at a blue vertex which is a vertex of degree 1 in the P₃ and is adjacent to a blue vertex and with the remaining vertex colored red, and (ii) F is a 2-stratified K₃ rooted at a blue vertex and with exactly one red vertex.     

Power domination in graphs

The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well-known domination problem in graphs. In 1998, Haynes et al. considered the graph theoretical representation of this problem as a variation of the domination problem. They defined a set S to be a power dominating set of a graph if every vertex and every edge in the system is monitored by the set S (following a set of rules for power system monitoring). The power domination number γ_P(G) of a graph G is the minimum cardinality of a power dominating set of G. In this paper, we present upper bounds on the power domination number for a connected graph with at least three vertices and a connected claw-free cubic graph in terms of their order. The extremal graphs attaining the upper bounds are also characterized.     

Changing and unchanging domination: a classification

The six classes of graphs resulting from the changing or unchanging of the domination number of a graph when a vertex is deleted, or an edge is deleted or added are considered. Each of these classes has been studied individually in the literature. We consider relationships among the classes, which are illustrated in a Venn diagram. We show that no subset of the Venn diagram is empty for arbitrary graphs, and prove that some of the subsets are empty for connected graphs. Our main result is a characterization of trees in each subset of the Venn diagram.     

On matching and total domination in graphs

A set M of edges of a graph G is a matching if no two edges in M are incident to the same vertex. A set S of vertices in G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The matching number is the maximum cardinality of a matching of G, while the total domination number of G is the minimum cardinality of a total dominating set of G. In this paper, we investigate the relationships between the matching and total domination number of a graph. We observe that the total domination number of every claw-free graph with minimum degree at least three is bounded above by its matching number, and we show that every k-regular graph with k?3 has total domination number at most its matching number. In general, we show that no minimum degree is sufficient to guarantee that the matching number and total domination number are comparable.     

Bounds on total domination in claw-free cubic graphs

A set S of vertices in a graph G is a total dominating set, denoted by TDS, of G if every vertex of G is adjacent to some vertex in S (other than itself). The minimum cardinality of a TDS of G is the total domination number of G, denoted by γ_t(G). If G does not contain K_1,3 as an induced subgraph, then G is said to be claw-free. It is shown in [D. Archdeacon, J. Ellis-Monaghan, D. Fischer, D. Froncek, P.C.B. Lam, S. Seager, B. Wei, R. Yuster, Some remarks on domination, J. Graph Theory 46 (2004) 207-210.] that if G is a graph of order n with minimum degree at least three, then γ_t(G)?n/2. Two infinite families of connected cubic graphs with total domination number one-half their orders are constructed in [O. Favaron, M.A. Henning, C.M. Mynhardt, J. Puech, Total domination in graphs with minimum degree three, J. Graph Theory 34(1) (2000) 9-19.] which shows that this bound of n/2 is sharp. However, every graph in these two families, except for K₄ and a cubic graph of order eight, contains a claw. It is therefore a natural question to ask whether this upper bound of n/2 can be improved if we restrict G to be a connected cubic claw-free graph of order at least 10. In this paper, we answer this question in the affirmative. We prove that if G is a connected claw-free cubic graph of order n?10, then γ_t(G)?5n/11.     

Distance paired domination numbers of graphs

In this paper, we study a generalization of the paired domination number. Let G=(V,E) be a graph without an isolated vertex. A set D⊆V(G) is a k-distance paired dominating set of G if D is a k-distance dominating set of G and the induced subgraph 〈D〉 has a perfect matching. The k-distance paired domination number is the cardinality of a smallest k-distance paired dominating set of G. We investigate properties of the k-distance paired domination number of a graph. We also give an upper bound and a lower bound on the k-distance paired domination number of a non-trivial tree T in terms of the size of T and the number of leaves in T and we also characterize the extremal trees.     

Total restrained domination numbers of trees

For a given connected graph G=(V,E), a set D_tr⊆V(G) is a total restrained dominating set if it is dominating and both 〈D_tr〉 and 〈V(G)-D_tr〉 do not contain isolate vertices. The cardinality of the minimum total restrained dominating set in G is the total restrained domination number and is denoted by γ_tr(G). In this paper we characterize the trees with equal total and total restrained dominating numbers and give a lower bound on the total restrained dominating number of a tree T in terms of its order and the number of leaves of T.     

Global italian domination in graphs

Abstract

An Italian dominating function (IDF) on a graph G = (V, E) is a function f: V → {0, 1, 2} satisfying the condition that for every vertex v ∈ V (G) with f (v) = 0, either v is adjacent to a vertex assigned 2 under f, or v is adjacent to at least two vertices assigned 1. The weight of an IDF f is the value ∑_v∈V(G) f (v). The Italian domination number of a graph G, denoted by γ_I (G), is the minimum weight of an IDF on G. An IDF f on G is called a global Italian dominating function (GIDF) on G if f is also an IDF on the complement ? of G. The global Italian domination number of G, denoted by γ_gI (G), is the minimum weight of a GIDF on G. In this paper, we initiate the study of the global Italian domination number and we present some strict bounds for the global Italian domination number. In particular, we prove that for any tree T of order n ≥ 4, γ_gI (T) ≤ γ_I (T) + 2 and we characterize all trees with γ_gI (T) = γ_I (T) + 2 and γ_gI (T) = γ_I (T) + 1.     

Domination versus disjunctive domination in graphs

A dominating set in a graph G is a set S of vertices of G such that every vertex not in S is adjacent to a vertex of S. The domination number of G is the minimum cardinality of a dominating set of G. For a positive integer b, a set S of vertices in a graph G is a b-disjunctive dominating set in G if every vertex v not in S is adjacent to a vertex of S or has at least b vertices in S at distance 2 from it in G. The b-disjunctive domination number of G is the minimum cardinality of a b-disjunctive dominating set. In this paper, we continue the study of disjunctive domination in graphs. We present properties of b-disjunctive dominating sets in a graph. A characterization of minimal b-disjunctive dominating sets is given. We obtain bounds on the ratio of the domination number and the b-disjunctive domination number for various families of graphs, including regular graphs and trees.     

Construction of trees and graphs with equal domination parameters

We provide a simple constructive characterization for trees with equal domination and independent domination numbers, and for trees with equal domination and total domination numbers. We also consider a general framework for constructive characterizations for other equality problems.     

Total domination subdivision numbers of trees

A set S of vertices in a graph G is a total dominating set of G if every vertex is adjacent to a 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_γt(G) of a graph G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the total domination number. Haynes et al. (J. Combin. Math. Combin. Comput. 44 (2003) 115) showed that for any tree T of order at least 3, 1?sd_γt(T)?3. In this paper, we give a constructive characterization of trees whose total domination subdivision number is 3.     

Distance irredundance and connected domination numbers of a graph

Let k be a positive integer and G be a connected graph. This paper considers the relations among four graph theoretical parameters: the k-domination number γ_k(G), the connected k-domination number ; the k-independent domination number and the k-irredundance number ir_k(G). The authors prove that if an ir_k-set X is a k-independent set of G, then , and that for k?2, if ir_k(G)=1, if ir_k(G) is odd, and if ir_k(G) is even, which generalize some known results.