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).     

Dominating direct products of graphs

An upper bound for the domination number of the direct product of graphs is proved. It in particular implies that for any graphs G and H, γ(G×H)?3γ(G)γ(H). Graphs with arbitrarily large domination numbers are constructed for which this bound is attained. Concerning the upper domination number we prove that Γ(G×H)?Γ(G)Γ(H), thus confirming a conjecture from [R. Nowakowski, D.F. Rall, Associative graph products and their independence, domination and coloring numbers, Discuss. Math. Graph Theory 16 (1996) 53-79]. Finally, for paired-domination of direct products we prove that γ_pr(G×H)?γ_pr(G)γ_pr(H) for arbitrary graphs G and H, and also present some infinite families of graphs that attain this bound.     

Lower bounds for the domination number and the total domination number of direct product graphs

A sharp lower bound for the domination number and the total domination number of the direct product of finitely many complete graphs is given: . Sharpness is established in the case when the factors are large enough in comparison to the number of factors. The main result gives a lower bound for the domination (and the total domination) number of the direct product of two arbitrary graphs: γ(G×H)≥γ(G)+γ(H)−1. Infinite families of graphs that attain the bound are presented. For these graphs it also holds that γ_t(G×H)=γ(G)+γ(H)−1. Some additional parallels with the total domination number are made.     

On the domination number of the cartesian product of the cycle of length n and any graph

Let γ(G) denote the domination number of a graph G and let C_n□G denote the cartesian product of C_n, the cycle of length n?3, and G. In this paper, we are mainly concerned with the question: which connected nontrivial graphs satisfy γ(C_n□G)=γ(C_n)γ(G)? We prove that this equality can only hold if n≡1 (mod 3). In addition, we characterize graphs which satisfy this equality when n=4 and provide infinite classes of graphs for general n≡1 (mod 3).     

Spectral bounds for the clique and independence numbers of graphs

We obtain a sequence k₁(G) ≤ k₂(G) ≤ … ≤ k_n(G) of lower bounds for the clique number (size of the largest clique) of a graph G of n vertices. The bounds involve the spectrum of the adjacency matrix of G. The bound k₁(G) is explicit and improves earlier known theorems. The bound k₂(G) is also explicit, and is shown to improve on the bound from Brooks' theorem even for regular graphs. The bounds k₃,…, k_r are polynomial-time computable, where r is the number of positive eigenvalues of G.     

A Characterization of 3-(γ c , 2)-Critical Claw-Free Graphs Which are not 3-γ c -Critical

Let γ _c (G) denote the minimum cardinality of a connected dominating set for G. A graph G is k-γ _c -critical if γ _c (G) = k, but γ _c (G + xy) < k for ${xy \in E(\overline {G})}$ . Further, for integer r ≥ 2, G is said to be k-(γ _c , r)-critical if γ _c (G) = k, but γ _c (G + xy) < k for each pair of non-adjacent vertices x and y that are at distance at most r apart. k-γ _c -critical graphs are k-(γ _c , r)-critical but the converse need not be true. In this paper, we give a characterization of 3-(γ _c , 2)-critical claw-free graphs which are not 3-γ _c -critical. In fact, we show that there are exactly four classes of such graphs.     

On the total domination number of cross products of graphs

We give lower and upper bounds on the total domination number of the cross product of two graphs, γ_t(G×H). These bounds are in terms of the total domination number and the maximum degree of the factors and are best possible. We further investigate cross products involving paths and cycles. We determine the exact values of γ_t(G×Pn) and γ_t(C_n×C_m) where P_n and C_n denote, respectively, a path and a cycle of length n.     

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.     

Properties of total domination edge-critical graphs

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γ_t(G) of G. The graph G is total domination edge critical if for every edge e in the complement of G, γ_t(G+e)<γ_t(G). We call such graphs γ_tEC. Properties of γ_tEC graphs are established.     

Nombre de representations d'une permutation comme produit de deux cycles de longueurs donnees

We consider the Ramsey number r(sK₂, G) where sK₂(s?1) denotes a set of s disjoint edges and G is an arbitrary finite simple graph with no isolated vertices. We obtain upper and lower bounds in the general case. Exact results are obtained for certain classes of graphs.     

Secure domination critical graphs

A secure dominating set X of a graph G is a dominating set with the property that each vertex u∈V_G−X is adjacent to a vertex v∈X such that (X−{v})∪{u} is dominating. The minimum cardinality of such a set is called the secure domination number, denoted by γ_s(G). We are interested in the effect of edge removal on γ_s(G), and characterize γ_s-ER-critical graphs, i.e. graphs for which γ_s(G−e)>γ_s(G) for any edge e of G, bipartite γ_s-ER-critical graphs and γ_s-ER-critical trees.     

The diameter of total domination vertex critical graphs

A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G-v is less than the total domination number of G. These graphs we call γ_t-critical. If such a graph G has total domination number k, we call it k-γ_t-critical. We characterize the connected graphs with minimum degree one that are γ_t-critical and we obtain sharp bounds on their maximum diameter. We calculate the maximum diameter of a k-γ_t-critical graph for k?8 and provide an example which shows that the maximum diameter is in general at least 5k/3-O(1).     

Domination critical graphs

A set of vertices S is said to dominate the graph G if for each v ? S, there is a vertex u ∈ S with u adjacent to v. The smallest cardinality of any such dominating set is called the domination number of G and is denoted by γ(G). The purpose of this paper is to initiate an investigation of those graphs which are critical in the following sense: For each v, u ∈ V(G) with v not adjacent to u, γ(G + vu) < γ(G). Thus G is k-y-critical if γ(G) = k and for each edge e ? E(G), γ(G + e) = k ?1. The 2-domination critical graphs are characterized the properties of the k-critical graphs with k ≥ 3 are studied. In particular, the connected 3-critical graphs of even order are shown to have a 1-factor and some stringent restrictions on their degree sequences and diameters are obtained.     

Extremal perfect graphs for a bound on the domination number

We examine classes of extremal graphs for the inequality γ(G)?|V|-max{d(v)+β_v(G)}, where γ(G) is the domination number of graph G, d(v) is the degree of vertex v, and β_v(G) is the size of a largest matching in the subgraph of G induced by the non-neighbours of v. This inequality improves on the classical upper bound |V|-maxd(v) due to Claude Berge. We give a characterization of the bipartite graphs and of the chordal graphs that achieve equality in the inequality. The characterization implies that the extremal bipartite graphs can be recognized in polynomial time, while the corresponding problem remains NP-complete for the extremal chordal graphs.     

On the packing chromatic number of Cartesian products, hexagonal lattice, and trees

The packing chromatic number χ_ρ(G) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into packings with pairwise different widths. Several lower and upper bounds are obtained for the packing chromatic number of Cartesian products of graphs. It is proved that the packing chromatic number of the infinite hexagonal lattice lies between 6 and 8. Optimal lower and upper bounds are proved for subdivision graphs. Trees are also considered and monotone colorings are introduced.     

Roman domination in regular graphs

A Roman domination function on a graph G=(V(G),E(G)) is a function f:V(G)→{0,1,2} satisfying the condition that every vertex u for which f(u)=0 is adjacent to at least one vertex v for which f(v)=2. The weight of a Roman dominating function is the value f(V(G))=∑_u∈V(G)f(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. Cockayne et al. [E. J. Cockayne et al. Roman domination in graphs, Discrete Mathematics 278 (2004) 11-22] showed that γ(G)≤γ_R(G)≤2γ(G) and defined a graph G to be Roman if γ_R(G)=2γ(G). In this article, the authors gave several classes of Roman graphs: P_3k,P_3k+2,C_3k,C_3k+2 for k≥1, K_m,n for min{m,n}≠2, and any graph G with γ(G)=1; In this paper, we research on regular Roman graphs and prove that: (1) the circulant graphs and , n⁄≡1 (mod (2k+1)), (n≠2k) are Roman graphs, (2) the generalized Petersen graphs P(n,2k+1)( (mod 4) and ), P(n,1) (n⁄≡2 (mod 4)), P(n,3) ( (mod 4)) and P(11,3) are Roman graphs, and (3) the Cartesian product graphs are Roman graphs.     

Sharp bounds on the distance spectral radius and the distance energy of graphs

The D-eigenvalues {μ₁,μ₂,…,…,μ_p} of a graph G are the eigenvalues of its distance matrix D and form the D-spectrum of G denoted by spec_D(G). The greatest D-eigenvalue is called the D-spectral radius of G denoted by μ₁. The D-energy E_D(G) of the graph G is the sum of the absolute values of its D-eigenvalues. In this paper we obtain some lower bounds for μ₁ and characterize those graphs for which these bounds are best possible. We also obtain an upperbound for E_D(G) and determine those maximal D-energy graphs.     

Sharp bounds for the largest eigenvalue of the signless Laplacian of a graph

Let G be a simple connected graph with n vertices and m edges. Denote the degree of vertex v_i by d(v_i). The matrix Q(G)=D(G)+A(G) is called the signless Laplacian of G, where D(G)=diag(d(v₁),d(v₂),…,d(v_n)) and A(G) denote the diagonal matrix of vertex degrees and the adjacency matrix of G, respectively. Let q₁(G) be the largest eigenvalue of Q(G). In this paper, we first present two sharp upper bounds for q₁(G) involving the maximum degree and the minimum degree of the vertices of G and give a new proving method on another sharp upper bound for q₁(G). Then we present three sharp lower bounds for q₁(G) involving the maximum degree and the minimum degree of the vertices of G. Moreover, we determine all extremal graphs which attain these sharp bounds.     

Connectivity measures in matched sum graphs

A matched sum graph G of two graphs G₁ and G₂ of the same order is obtained from the union of G₁ and G₂ and from joining each vertex of G₁ with one vertex of G₂ according to one bijection f between the vertices in V(G₁) and V(G₂). When G₁=G₂=H then f is just a permutation of V(H) and the corresponding matched sum graph is a permutation graph H^f. In this paper, we derive lower bounds for the connectivity, edge-connectivity, and different conditional connectivities in matched sum graphs, and present sufficient conditions which guarantee maximum values for these conditional connectivities.     

On Edge Semi-Isomorphisms and Semi-Dualities of Graphs

Let G₁ and G₂ be undirected graphs, and ?₁(G ₁) and ?₂(G ₂) be families of edge sets of G₁ and G₂, respectively. An (?₁,?₂)-semi-isomorphism ofG ₁ ontoG ₂ is an edge bijection between G₁ and G₂ that induces an injection from ?₁(G ₁) to ?₂(G ₂). This concept generalizes a well known concept of a circuit isomorphism of graphs due to H. Whitney. If has a “dual nature” with respect to ?₂(G ₂) then the concept of (?₁,?₂)-semi-isomorphism of graphs turns into a concept of a (?₁,?₂)-semi-duality of graphs. This gives a natural generalization of the circuit duality of graphs due to H. Whitney. In this paper we investigate (?₁,?₂)-semi-isomorphisms and (?₁,?₂)-semi-dualities of graphs for various families ?₁(G ₁) and ?₂(G ₂). In particular, we consider families of circuits and cocircuits of graphs from this point of view, and obtain some strengthenings of Whitney’s 2-isomorphism theorem and Whitney’s planarity criterion for 3-connected graphs.