On a conjecture of Fink and Jacobson concerning k-domination and k-dependence

A set D of vertices of a graph is k-dependent if every vertex of D is joined to at most k?1 vertices in D. Let β_k(G) be the maximum order of a k-dependent set in G. A set D of vertices of G is k-dominating if every vertex not in D is joined to at least k vertices of D. Let γ_k(G) be the minimum order of a k-dominating set in G. Here we prove the following conjecture of Fink and Jacobson: for any simple graph G and any positive integer k, γ_k(G) ≤ β_k(G).     

Some graft transformations and its applications on the distance spectral radius of a graph

Let D(G)=(d_i,j)_n×n denote the distance matrix of a connected graph G with order n, where d_ij is equal to the distance between v_i and v_j in G. The largest eigenvalue of D(G) is called the distance spectral radius of graph G, denoted by ?(G). In this paper, some graft transformations that decrease or increase ?(G) are given. With them, for the graphs with both order n and k pendant vertices, the extremal graphs with the minimum distance spectral radius are completely characterized; the extremal graph with the maximum distance spectral radius is shown to be a dumbbell graph (obtained by attaching some pendant edges to each pendant vertex of a path respectively) when 2≤k≤n−2; for k=1,2,3,n−1, the extremal graphs with the maximum distance spectral radius are completely characterized.     

The ratio of the distance irredundance and domination numbers of a graph

Let n ≥ 1 be an integer and let G be a graph. A set D of vertices in G is defined to be an n-dominating set of G if every vertex of G is within distance n from some vertex of D. The minimum cardinality among all n-dominating sets of G is called the n-domination number of G and is denoted by γ_n(G). A set / of vertices in G is n-irredundant if for every vertex x ∈ / there exists a vertex y that is within distance n from x but at distance greater than n from every vertex of / - {x}. The n-irredundance number of G, denoted by ir_n(G), is the minimum cardinality taken over all maximal n-irredundant sets of vertices of G. We show that inf{ir_n(G)/γ_n(G) | G is an arbitrary finite undirected graph with neither loops nor multiple edges} = 1/2 with the infimum not being attained. Subsequently, we show that 2/3 is a lower bound on all quotients ir_n(T)/γ_n(T) in which T is a tree. Furthermore, we show that, for n ≥ 2, this bound is sharp. These results extend those of R. B. Allan and R.C. Laskar [“On Domination and Some Related Concepts in Graph Theory,” Utilitas Mathematica, Vol. 21 (1978), pp. 43–56], B. Bollobás and E. J. Cockayne [“Graph-Theoretic Parameters Concerning Domination, Independence and Irredundance,” Journal of Graph Theory, Vol. 3 (1979), pp. 241–249], and P. Damaschke [Irredundance Number versus Domination Number, Discrete Mathematics, Vol. 89 (1991), pp. 101–104].     

Distance graphs with maximum chromatic number

The distance graph G(D) has the set of integers as vertices and two vertices are adjacent in G(D) if their difference is contained in the set D⊂Z. A conjecture of Zhu states that if the chromatic number of G(D) achieves its maximum value |D|+1 then the graph has a triangle. The conjecture is proven to be true if |D|?3. We prove that the chromatic number of a distance graph with D={a,b,c,d} is five only if either D={1,2,3,4k} or D={a,b,a+b,b-a}. This confirms a stronger version of Zhu's conjecture for |D|=4, namely, if the chromatic number achieves its maximum value then the graph contains K₄.     

Connected graphs with maximal Q-index: The one-dominating-vertex case

By the signless Laplacian of a (simple) graph G we mean the matrix Q(G)=D(G)+A(G), where A(G),D(G) denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of G. For every pair of positive integers n,k, it is proved that if 3?k?n-3, then H_n,k, the graph obtained from the star K_1,n-1 by joining a vertex of degree 1 to k+1 other vertices of degree 1, is the unique connected graph that maximizes the largest signless Laplacian eigenvalue over all connected graphs with n vertices and n+k edges.     

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.     

The <Emphasis Type="Italic">k</Emphasis>-domatic number of a graph

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A k-dominating set of the graph G is a subset D of V (G) such that every vertex of V (G)-D is adjacent to at least k vertices in D. A k-domatic partition of G is a partition of V (G) into k-dominating sets. The maximum number of dominating sets in a k-domatic partition of G is called the k-domatic number d _k (G). In this paper, we present upper and lower bounds for the k-domatic number, and we establish Nordhaus-Gaddum-type results. Some of our results extend those for the classical domatic number d(G) = d ₁(G).      

The k-Dominating Graph

Given a graph G, the k-dominating graph of G, D _k (G), is defined to be the graph whose vertices correspond to the dominating sets of G that have cardinality at most k. Two vertices in D _k (G) are adjacent if and only if the corresponding dominating sets of G differ by either adding or deleting a single vertex. The graph D _k (G) aids in studying the reconfiguration problem for dominating sets. In particular, one dominating set can be reconfigured to another by a sequence of single vertex additions and deletions, such that the intermediate set of vertices at each step is a dominating set if and only if they are in the same connected component of D _k (G). In this paper we give conditions that ensure D _k (G) is connected.     

Coloring of distance graphs with intervals as distance sets

Let D be a set of positive integers. The distance graph G(Z,D) with distance set D is the graph with vertex set Z in which two vertices x,y are adjacent if and only if |x−y|D. The fractional chromatic number, the chromatic number, and the circular chromatic number of G(Z,D) for various D have been extensively studied recently. In this paper, we investigate the fractional chromatic number, the chromatic number, and the circular chromatic number of the distance graphs with the distance sets of the form D_m,[k,k^′]={1,2,…,m}−{k,k+1,…,k^′}, where m, k, and k^′ are natural numbers with m≥k^′≥k. In particular, we completely determine the chromatic number of G(Z,D_m,[2,k^′]) for arbitrary m, and k^′.     

On graphs with equal domination and 2-domination numbers

Let G be a simple graph, and let p be a positive integer. A subset D⊆V(G) is a p-dominating set of the graph G, if every vertex v∈V(G)-D is adjacent to at least p vertices in D. The p-domination numberγ_p(G) is the minimum cardinality among the p-dominating sets of G. Note that the 1-domination number γ₁(G) is the usual domination numberγ(G). This definition immediately leads to the inequality γ(G)?γ₂(G).In this paper we present some sufficient as well as some necessary conditions for graphs G with the property that γ₂(G)=γ(G). In particular, we characterize all cactus graphs H with γ₂(H)=γ(H).     

An upper bound on the domination number of a graph with minimum degree 2

A set S of vertices in a graph G is a dominating set of G if every vertex of V(G)?S is adjacent to some vertex in S. The minimum cardinality of a dominating set of G is the domination number of G, denoted as γ(G). Let P_n and C_n denote a path and a cycle, respectively, on n vertices. Let k₁(F) and k₂(F) denote the number of components of a graph F that are isomorphic to a graph in the family {P₃,P₄,P₅,C₅} and {P₁,P₂}, respectively. Let L be the set of vertices of G of degree more than 2, and let G−L be the graph obtained from G by deleting the vertices in L and all edges incident with L. McCuaig and Shepherd [W. McCuaig, B. Shepherd, Domination in graphs with minimum degree two, J. Graph Theory 13 (1989) 749-762] showed that if G is a connected graph of order n≥8 with δ(G)≥2, then γ(G)≤2n/5, while Reed [B.A. Reed, Paths, stars and the number three, Combin. Probab. Comput. 5 (1996) 277-295] showed that if G is a graph of order n with δ(G)≥3, then γ(G)≤3n/8. As an application of Reed’s result, we show that if G is a graph of order n≥14 with δ(G)≥2, then .     

Distance domination of generalized de Bruijn and Kautz digraphs

Let G = (V,A) be a digraph and k ≥ 1 an integer. For u, v ∈ V, we say that the vertex u distance k-dominate v if the distance from u to v at most k. A set D of vertices in G is a distance k-dominating set if each vertex of V D is distance k-dominated by some vertex of D. The distance k-domination number of G, denoted by γ _k (G), is the minimum cardinality of a distance k-dominating set of G. Generalized de Bruijn digraphs G _B (n, d) and generalized Kautz digraphs G _K (n, d) are good candidates for interconnection networks. Denote Δ _k := (∑ _j=0 ^k d ^j )^?1. F. Tian and J. Xu showed that ?nΔ _k ? γ _k (G _B (n, d)) ≤?n/d ^k? and ?nΔ _k ? ≤ γ _k (G _K (n, d)) ≤ ?n/d ^k ?. In this paper, we prove that every generalized de Bruijn digraph G _B (n, d) has the distance k-domination number ?nΔ _k ? or ?nΔ _k ?+1, and the distance k-domination number of every generalized Kautz digraph G _K (n, d) bounded above by ?n/(d ^k?1+d ^k )?. Additionally, we present various sufficient conditions for γ _k (G _B (n, d)) = ?nΔ _k ? and γ _k (G _K (n, d)) = ?nΔ _k ?.     

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.     

A new domination conception

Let f be an integer valued function defined on the vertex set V(G) of a simple graph G. We call a subset D_f of V(G) a f-dominating set of G if |N(x, G) ∩ D_f| ≥ f(x) for all x ∈ V(G) — D_f, where N(x, G) is the set of neighbors of x. D_f is a minimum f-dominating set if G has no f-dominating set D′_f with |D_f| < |D_f|. If j, k ∈ N₀ = {0,1,2,…} with j ≤ k, then we define the integer valued function f_j,k on V(G) by . By μ_j,k(G) we denote the cardinality of a minimum f_j,k-dominating set of G. A set D ? V(G) is j-dominating if every vertex, which is not in D, is adjacent to at least j vertices of D. The j-domination number γ_j(G) is the minimum order of a j-dominating set in G. In this paper we shall give estimations of the new domination number μ_j,k(G), and with the help of these estimations we prove some new and some known upper bounds for the j-domination number. © 1993 John Wiley & Sons, Inc.     

The vertex linear arboricity of distance graphs

A distance graph is a graph G(R,D) with the set of all points of the real line as vertex set and two vertices u,v∈R are adjacent if and only if |u-v|∈D where the distance set D is a subset of the positive real numbers. Here, the vertex linear arboricity of G(R,D) is determined when D is an interval between 1 and δ. In particular, the vertex linear arboricity of integer distance graphs G(D) is discussed, too.     

Matchings in 3-vertex-critical graphs: The odd case

A subset of vertices D of a graph G is a dominating set for G if every vertex of G not in D is adjacent to one in D. The cardinality of any smallest dominating set in G is denoted by γ(G) and called the domination number of G. Graph G is said to be γ-vertex-critical if γ(G-v)<γ(G), for every vertex v in G. A graph G is said to be factor-critical if G-v has a perfect matching for every choice of v∈V(G).In this paper, we present two main results about 3-vertex-critical graphs of odd order. First we show that any such graph with positive minimum degree and at least 11 vertices which has no induced subgraph isomorphic to the bipartite graph K_1,5 must contain a near-perfect matching. Secondly, we show that any such graph with minimum degree at least three which has no induced subgraph isomorphic to the bipartite graph K_1,4 must be factor-critical. We then show that these results are best possible in several senses and close with a conjecture.