A note on Roman domination in 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 of G, γ(G), equals the minimum cardinality of a dominating set. A Roman dominating function on graph G=(V,E) is a function f:V→{0,1,2} satisfying the condition that every vertex v for which f(v)=0 is adjacent to at least one vertex u for which f(u)=2. The weight of a Roman dominating function is the value f(V)=∑_v∈Vf(v). The Roman domination number of a graph G, denoted by γ_R(G), equals the minimum weight of a Roman dominating function on G. In this paper, for any integer k(2?k?γ(G)), we give a characterization of graphs for which γ_R(G)=γ(G)+k, which settles an open problem in [E.J. Cockayne, P.M. Dreyer Jr, S.M. Hedetniemi et al. On Roman domination in graphs, Discrete Math. 278 (2004) 11-22].     

Signed star domatic number of a graph

Let G be a simple graph without isolated vertices with vertex set V(G) and edge set E(G). A function f:E(G)?{−1,1} is said to be a signed star dominating function on G if ∑_e∈E(v)f(e)≥1 for every vertex v of G, where E(v)={uv∈E(G)∣u∈N(v)}. A set {f₁,f₂,…,f_d} of signed star dominating functions on G with the property that for each e∈E(G), is called a signed star dominating family (of functions) on G. The maximum number of functions in a signed star dominating family on G is the signed star domatic number of G, denoted by d_SS(G).In this paper we study the properties of the signed star domatic number d_SS(G). In particular, we determine the signed domatic number of some classes of graphs.     

Factor-critical property in 3-dominating-critical graphs

A vertex subset S of a graph G is a dominating set if every vertex of G either belongs to S or is adjacent to a vertex of S. The cardinality of a smallest dominating set is called the dominating number of G and is denoted by γ(G). A graph G is said to be γ-vertex-critical if γ(G−v)<γ(G), for every vertex v in G.Let G be a 2-connected K_1,5-free 3-vertex-critical graph of odd order. For any vertex v∈V(G), we show that G−v has a perfect matching (except two graphs), which solves a conjecture posed by Ananchuen and Plummer [N. Ananchuen, M.D. Plummer, Matchings in 3-vertex critical graphs: The odd case, Discrete Math., 307 (2007) 1651-1658].     

On the eccentric distance sum of trees and unicyclic graphs

Let G be a simple connected graph with the vertex set V(G). The eccentric distance sum of G is defined as ξd(G)=_∑v∈V(G)ε(v)DG(v), where ε(v) is the eccentricity of the vertex v and DG(v)=_∑u∈V(G)d(u,v) is the sum of all distances from the vertex v. In this paper we characterize the extremal unicyclic graphs among n-vertex unicyclic graphs with given girth having the minimal and second minimal eccentric distance sum. In addition, we characterize the extremal trees with given diameter and minimal eccentric distance sum.     

A constructive characterization of total domination vertex critical graphs

Let G be a graph of order n and maximum degree Δ(G) and let γ_t(G) denote the minimum cardinality of a total dominating set of a graph G. A graph G with no isolated vertex is the 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. We call these graphs γ_t-critical. For any γ_t-critical graph G, it can be shown that n≤Δ(G)(γ_t(G)−1)+1. In this paper, we prove that: Let G be a connected γ_t-critical graph of order n (n≥3), then n=Δ(G)(γ_t(G)−1)+1 if and only if G is regular and, for each v∈V(G), there is an A⊆V(G)−{v} such that N(v)∩A=0?, the subgraph induced by A is 1-regular, and every vertex in V(G)−A−{v} has exactly one neighbor in A.     

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.     

Domination with exponential decay

Let G be a graph and S⊆V(G). For each vertex u∈S and for each v∈V(G)−S, we define to be the length of a shortest path in 〈V(G)−(S−{u})〉 if such a path exists, and ∞ otherwise. Let v∈V(G). We define if v⁄∈S, and w_S(v)=2 if v∈S. If, for each v∈V(G), we have w_S(v)≥1, then S is an exponential dominating set. The smallest cardinality of an exponential dominating set is the exponential domination number, γ_e(G). In this paper, we prove: (i) that if G is a connected graph of diameter d, then γ_e(G)≥(d+2)/4, and, (ii) that if G is a connected graph of order n, then .     

On minimum secure dominating sets of graphs

A subset X of the vertex set of a graph G is a secure dominating set of G if X is a dominating set of G and if, for each vertex u not in X, there is a neighbouring vertex v of u in X such that the swap set (X/{v}) ? {u} is again a dominating set of G, in which case v is called a defender. The secure domination number of G is the cardinality of a smallest secure dominating set of G. In this paper, we show that every graph of minimum degree at least 2 possesses a minimum secure dominating set in which all vertices are defenders. We also characterise the classes of graphs that have secure domination numbers 1, 2 and 3.     

On distance two graphs of upper bound graphs

For a poset P=(X,≤), the upper bound graph (UB-graph) of P is the graph U=(X,E_U), where uv∈E_U if and only if u≠v and there exists m∈X such that u,v≤m. For a graph G, the distance two graph DS₂(G) is the graph with vertex set V(DS₂(G))=V(G) and u,v∈V(DS₂(G)) are adjacent if and only if d_G(u,v)=2. In this paper, we deal with distance two graphs of upper bound graphs. We obtain a characterization of distance two graphs of split upper bound graphs.     

Regular graphs are not universal fixers

For any permutation π of the vertex set of a graph G, the graph πG is obtained from two copies G^′ and G^″ of G by joining u∈V(G^′) and v∈V(G^″) if and only if v=π(u). Denote the domination number of G by γ(G). For all permutations π of V(G), γ(G)≤γ(πG)≤2γ(G). If γ(πG)=γ(G) for all π, then G is called a universal fixer. We prove that regular graphs and graphs with γ=4 are not universal fixers.     

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.     

The signed domatic number of some regular graphs

Let G be a finite and simple graph with vertex set V(G), and let f:V(G)→{−1,1} be a two-valued function. If ∑_x∈N[v]f(x)≥1 for each v∈V(G), where N[v] is the closed neighborhood of v, then f is a signed dominating function on G. A set {f₁,f₂,…,f_d} of signed dominating functions on G with the property that for each x∈V(G), is called a signed dominating family (of functions) on G. The maximum number of functions in a signed dominating family on G is the signed domatic number on G. In this paper, we investigate the signed domatic number of some circulant graphs and of the torus C_p×C_q.     

Connected domination of regular graphs

A dominating setD of a graph G is a subset of V(G) such that for every vertex v∈V(G), either v∈D or there exists a vertex u∈D that is adjacent to v in G. Dominating sets of small cardinality are of interest. A connected dominating setC of a graph G is a dominating set of G such that the subgraph induced by the vertices of C in G is connected. A weakly-connected dominating setW of a graph G is a dominating set of G such that the subgraph consisting of V(G) and all edges incident with vertices in W is connected. In this paper we present several algorithms for finding small connected dominating sets and small weakly-connected dominating sets of regular graphs. We analyse the average-case performance of these heuristics on random regular graphs using differential equations, thus giving upper bounds on the size of a smallest connected dominating set and the size of a smallest weakly-connected dominating set of random regular graphs.     

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.     

On the extendability of Bi-Cayley graphs of finite abelian groups

Let G be a finite group and A a nonempty subset (possibly containing the identity element) of G. The Bi-Cayley graph X=BC(G,A) of G with respect to A is defined as the bipartite graph with vertex set G×{0,1} and edge set {{(g,0),(sg,1)}∣g∈G,s∈A}. A graph Γ admitting a perfect matching is called n-extendable if ∣V(Γ)∣≥2n+2 and every matching of size n in Γ can be extended to a perfect matching of Γ. In this paper, the extendability of Bi-Cayley graphs of finite abelian groups is explored. In particular, 2-extendable and 3-extendable Bi-Cayley graphs of finite abelian groups are characterized.     

Unique response Roman domination in graphs

A function f:V(G)→{0,1,2} is a Roman dominating function if every vertex u for which f(u)=0 is adjacent to at least one vertex v for which f(v)=2. A function f:V(G)→{0,1,2} with the ordered partition (V₀,V₁,V₂) of V(G), where V_i={v∈V(G)∣f(v)=i} for i=0,1,2, is a unique response Roman function if x∈V₀ implies |N(x)∩V₂|≤1 and x∈V₁∪V₂ implies that |N(x)∩V₂|=0. A function f:V(G)→{0,1,2} is a unique response Roman dominating function if it is a unique response Roman function and a Roman dominating function. The unique response Roman domination number of G, denoted by u_R(G), is the minimum weight of a unique response Roman dominating function. In this paper we study the unique response Roman domination number of graphs and present bounds for this parameter.     

Global secure sets of grid-like graphs

Let G=(V,E) be a graph and S⊆V. The set S is a secure set if ∀X⊆S,|N[X]∩S|≥|N[X]−S|, and S is a global secure set if S is a secure set and a dominating set. The cardinality of a minimum global secure set of G is the global security number of G, denoted γ_s(G). The sets studied in this paper are different from secure dominating sets studied in Cockayne et al. (2003) [3], Grobler and Mynhardt (2009) [8], or Klostermeyer and Mynhardt (2008) [13], which are also denoted by γ_s.In this paper, we provide results on the global security numbers of paths, cycles and their Cartesian products.     

The signed star domination numbers of the Cartesian product graphs

Let G be a graph with vertex set V(G) and edge set E(G). A function f:E(G)→{-1,1} is said to be a signed star dominating function of G if for every v∈V(G), where E_G(v)={uv∈E(G)|u∈V(G)}. The minimum of the values of , taken over all signed star dominating functions f on G, is called the signed star domination number of G and is denoted by γ_SS(G). In this paper, a sharp upper bound of γ_SS(G×H) is presented.     

Rainbow domination on trees

This paper studies a variation of domination in graphs called rainbow domination. For a positive integer k, a k-rainbow dominating function of a graph G is a function f from V(G) to the set of all subsets of {1,2,…,k} such that for any vertex v with f(v)=0? we have ∪_u∈NG(v)f(u)={1,2,…,k}. The 1-rainbow domination is the same as the ordinary domination. The k-rainbow domination problem is to determine the k-rainbow domination number of a graph G, that is the minimum value of ∑_v∈V(G)|f(v)| where f runs over all k-rainbow dominating functions of G. In this paper, we prove that the k-rainbow domination problem is NP-complete even when restricted to chordal graphs or bipartite graphs. We then give a linear-time algorithm for the k-rainbow domination problem on trees. For a given tree T, we also determine the smallest k such that .