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 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.     

An Improved Upper Bound on the Total Restrained Domination Number in Cubic Graphs

In this paper, we continue the study of total restrained domination in graphs. A set S of vertices in a graph G = (V, E) is a total restrained dominating set of G if every vertex of G is adjacent to some vertex in S and every vertex of ${V {\setminus} S}$ is adjacent to a vertex in ${V {\setminus} S}$ . The minimum cardinality of a total restrained dominating set of G is the total restrained domination number γ _tr(G) of G. Jiang et?al. (Graphs Combin 25:341–350, 2009) showed that if G is a connected cubic graph of order n, then γ _tr(G) ≤ 13n/19. In this paper we improve this upper bound to γ _tr(G) ≤ (n?+?4)/2. We provide two infinite families of connected cubic graphs G with γ _tr(G) = n/2, showing that our new improved bound is essentially best possible.     

Characterization of double domination subdivision number of trees

In a graph G, a vertex dominates itself and its neighbors. A subset S⊆V(G) is a double dominating set of G if S dominates every vertex of G at least twice. The double domination numberdd(G) is the minimum cardinality of a double dominating set of G. The double domination subdivision numbersd_dd(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 double domination number. In this paper first we establish upper bounds on the double domination subdivision number for arbitrary graphs in terms of vertex degree. Then we present several different conditions on G which are sufficient to imply that sd_dd(G)?3. We also prove that 1?sd_dd(T)?2 for every tree T, and characterize the trees T for which sd_dd(T)=2.     

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.     

On equality in an upper bound for the restrained and total domination numbers of a graph

Let G=(V,E) be a graph. A set S⊆V is a restrained dominating set (RDS) 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 an RDS of G. A set S⊆V is a total dominating set (TDS) if every vertex in V is adjacent to a vertex in S. The total domination number of a graph G without isolated vertices, denoted by γ_t(G), is the minimum cardinality of a TDS of G.Let δ and Δ denote the minimum and maximum degrees, respectively, in G. If G is a graph of order n with δ?2, then it is shown that γ_r(G)?n-Δ, and we characterize the connected graphs with δ?2 achieving this bound that have no 3-cycle as well as those connected graphs with δ?2 that have neither a 3-cycle nor a 5-cycle. Cockayne et al. [Total domination in graphs, Networks 10 (1980) 211-219] showed that if G is a connected graph of order n?3 and Δ?n-2, then γ_t(G)?n-Δ. We further characterize the connected graphs G of order n?3 with Δ?n-2 that have no 3-cycle and achieve γ_t(G)=n-Δ.     

Nordhaus-Gaddum results for restrained domination and total restrained domination in graphs

Let G=(V,E) be a graph. A set S⊆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. A set S⊆V is a restrained dominating set if every vertex in V-S is adjacent to a vertex in S and to a vertex in V-S. The total restrained domination number of G (restrained domination number of G, respectively), denoted by γ_tr(G) (γ_r(G), respectively), is the smallest cardinality of a total restrained dominating set (restrained dominating set, respectively) of G. We bound the sum of the total restrained domination numbers of a graph and its complement, and provide characterizations of the extremal graphs achieving these bounds. It is known (see [G.S. Domke, J.H. Hattingh, S.T. Hedetniemi, R.C. Laskar, L.R. Markus, Restrained domination in graphs, Discrete Math. 203 (1999) 61-69.]) that if G is a graph of order n?2 such that both G and are not isomorphic to P₃, then . We also provide characterizations of the extremal graphs G of order n achieving these bounds.     

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.     

Spectral radius of digraphs with given dichromatic number

Let D be a digraph with vertex set V(D). A partition of V(D) into k acyclic sets is called a k-coloring of D. The minimum integer k for which there exists a k-coloring of D is the dichromatic number χ(D) of the digraph D. Denote G_n,k the set of the digraphs of order n with the dichromatic number k≥2. In this note, we characterize the digraph which has the maximal spectral radius in G_n,k. Our result generalizes the result of [8] by Feng et al.     

Roman and Total Domination

Abstract

A set S of vertices is a total dominating set of a graph G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set is the total domination number ^γt(G). A Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u with f (u)=0 is adjacent to at least one vertex v of G for which f (v)=2. The minimum of f (V (G))=∑_{u ∈ V (G)} f (u) over all such functions is called the Roman domination number γR (G). We show that γt(G) ≤ γR (G) with equality if and only if ^γt(G)=2γ(G), where γ(G) is the domination number of G. Moreover, we characterize the extremal graphs for some graph families.     

An Upper Bound for the Total Restrained Domination Number of Graphs

Let G be a graph with vertex set V. A set ${D \subseteq V}$ is a total restrained dominating set of G if every vertex in V has a neighbor in D and every vertex in ${V \setminus D}$ has a neighbor in ${V \setminus D}$ . The minimum cardinality of a total restrained dominating set of G is called the total restrained domination number of G, and is denoted by γ _tr (G). In this paper, we prove that if G is a connected graph of order n ≥ 4 and minimum degree at least two, then ${\gamma_{tr}(G) \leq n-\sqrt[3]{n \over 4}}$ .     

Total restrained domination in trees

Let G=(V,E) be a graph. A set S⊆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 T is a tree of order n, then . Moreover, we show that if T is a tree of order , then . We then constructively characterize the extremal trees T of order n achieving these lower bounds.     

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.     

Total Restrained Domination in Cubic Graphs

A set S of vertices in a graph G = (V, E) is a total restrained dominating set (TRDS) of G if every vertex of G 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 minimum cardinality of a TRDS of G. Let G be a cubic graph of order n. In this paper we establish an upper bound on γ _tr (G). If adding the restriction that G is claw-free, then we show that γ _tr (G) = γ _t (G) where γ _t (G) is the total domination number of G, and thus some results on total domination in claw-free cubic graphs are valid for total restrained domination. Research was partially supported by the NNSF of China (Nos. 60773078, 10832006), the ShuGuang Plan of Shanghai Education Development Foundation (No. 06SG42) and Shanghai Leading Academic Discipline Project (No. S30104).     

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.     

Extremal problems on detectable colorings of trees

Let G be a connected graph of order 3 or more and let be a coloring of the edges of G (where adjacent edges may be colored the same). For each vertex v of G, the color code of v is the k-tuple c(v)=(a₁,a₂,…,a_k), where a_i is the number of edges incident with v that are colored i (1?i?k). The coloring c is called detectable if distinct vertices have distinct color codes; while the detection number det(G) of G is the minimum positive integer k for which G has a detectable k-coloring. For each integer n?3, let D_T(n) be the maximum detection number among all trees of order n and d_T(n) the minimum detection number among all trees of order n. The numbers D_T(n) and d_T(n) are determined for all integers n?3. Furthermore, it is shown that for integers k?2 and n?3, there exists a tree T of order n having det(T)=k if and only if d_T(n)?k?D_T(n).     

Upper minus total domination in small-degree regular graphs

A function f:V(G)→{-1,0,1} defined on the vertices of a graph G is a minus total dominating function (MTDF) if the sum of its function values over any open neighborhood is at least one. An MTDF f is minimal if there does not exist an MTDF g:V(G)→{-1,0,1}, f≠g, for which g(v)?f(v) for every v∈V(G). The weight of an MTDF is the sum of its function values over all vertices. The minus total domination number of G is the minimum weight of an MTDF on G, while the upper minus domination number of G is the maximum weight of a minimal MTDF on G. In this paper we present upper bounds on the upper minus total domination number of a cubic graph and a 4-regular graph and characterize the regular graphs attaining these upper bounds.     

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)).     

Maximum nullity of outerplanar graphs and the path cover number

Let G=(V,E) be a graph with V={1,2,…,n}. Define S(G) as the set of all n×n real-valued symmetric matrices A=[a_ij] with a_ij≠0,i≠j if and only if ij∈E. By M(G) we denote the largest possible nullity of any matrix A∈S(G). The path cover number of a graph G, denoted P(G), is the minimum number of vertex disjoint paths occurring as induced subgraphs of G which cover all the vertices of G.There has been some success with relating the path cover number of a graph to its maximum nullity. Johnson and Duarte [5], have shown that for a tree T,M(T)=P(T). Barioli et al. [2], show that for a unicyclic graph G,M(G)=P(G) or M(G)=P(G)-1. Notice that both families of graphs are outerplanar. We show that for any outerplanar graph G,M(G)?P(G). Further we show that for any partial 2-path G,M(G)=P(G).     

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.