A note on power domination in grid 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 vertex covering and dominating set problems in graphs (see [T.W. Haynes, S.M. Hedetniemi, S.T. Hedetniemi, M.A. Henning, Power domination in graphs applied to electrical power networks, SIAM J. Discrete Math. 15(4) (2002) 519-529]). A set S of vertices is defined 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 minimum cardinality of a power dominating set of a graph is its power domination number. In this paper, we determine the power domination number of an n×m grid graph.     

Total domination stable graphs upon edge addition

A set S of vertices in a graph G is a total dominating set 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 of G. A graph is total domination edge addition stable if the addition of an arbitrary edge has no effect on the total domination number. In this paper, we characterize total domination edge addition stable graphs. We determine a sharp upper bound on the total domination number of total domination edge addition stable graphs, and we determine which combinations of order and total domination number are attainable. We finish this work with an investigation of claw-free total domination edge addition stable graphs.     

A sharp upper bound on algebraic connectivity using domination number

Let G be a connected graph of order n. The algebraic connectivity of G is the second smallest eigenvalue of the Laplacian matrix of G. A dominating set in G is a vertex subset S such that each vertex of G that is not in S is adjacent to a vertex in S. The least cardinality of a dominating set is the domination number. In this paper, we prove a sharp upper bound on the algebraic connectivity of a connected graph in terms of the domination number and characterize the associated extremal graphs.     

Coloring, location and domination of corona graphs

A vertex coloring of a graph G is an assignment of colors to the vertices of G so that every two adjacent vertices of G have different colors. A coloring related property of a graphs is also an assignment of colors or labels to the vertices of a graph, in which the process of labeling is done according to an extra condition. A set S of vertices of a graph G is a dominating set in G if every vertex outside of S is adjacent to at least one vertex belonging to S. A domination parameter of G is related to those structures of a graph that satisfy some domination property together with other conditions on the vertices of G. In this article we study several mathematical properties related to coloring, domination and location of corona graphs. We investigate the distance-k colorings of corona graphs. Particularly, we obtain tight bounds for the distance-2 chromatic number and distance-3 chromatic number of corona graphs, through some relationships between the distance-k chromatic number of corona graphs and the distance-k chromatic number of its factors. Moreover, we give the exact value of the distance-k chromatic number of the corona of a path and an arbitrary graph. On the other hand, we obtain bounds for the Roman dominating number and the locating–domination number of corona graphs. We give closed formulaes for the k-domination number, the distance-k domination number, the independence domination number, the domatic number and the idomatic number of corona graphs.     

Total domination critical and stable graphs upon edge removal

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 of G. A graph is total domination edge critical if the removal of any arbitrary edge increases the total domination number. On the other hand, a graph is total domination edge stable if the removal of any arbitrary edge has no effect on the total domination number. In this paper, we characterize total domination edge critical graphs. We also investigate various properties of total domination edge stable graphs.     

Total domination dot-stable graphs

A set S of vertices in a graph G is a total dominating set 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 of G. Two vertices of G are said to be dotted (identified) if they are combined to form one vertex whose open neighborhood is the union of their neighborhoods minus themselves. We note that dotting any pair of vertices cannot increase the total domination number. Further we show it can decrease the total domination number by at most 2. A graph is total domination dot-stable if dotting any pair of adjacent vertices leaves the total domination number unchanged. We characterize the total domination dot-stable graphs and give a sharp upper bound on their total domination number. We also characterize the graphs attaining this bound.     

Total domination changing and stable graphs upon vertex removal

A set S of vertices in a graph G is a total dominating set 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 of G. A graph is total domination vertex removal stable if the removal of an arbitrary vertex leaves the total domination number unchanged. On the other hand, a graph is total domination vertex removal changing if the removal of an arbitrary vertex changes the total domination number. In this paper, we study total domination vertex removal changing and stable graphs.     

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.     

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 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.     

Total domination in 2‐connected graphs and in graphs with no induced 6‐cycles

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. It is known [J Graph Theory 35 (2000), 21–45] that if G is a connected graph of order n > 10 with minimum degree at least 2, then γ_t(G) ≤ 4n/7 and the (infinite family of) graphs of large order that achieve equality in this bound are characterized. In this article, we improve this upper bound of 4n/7 for 2‐connected graphs, as well as for connected graphs with no induced 6‐cycle. We prove that if G is a 2‐connected graph of order n > 18, then γ_t(G) ≤ 6n/11. Our proof is an interplay between graph theory and transversals in hypergraphs. We also prove that if G is a connected graph of order n > 18 with minimum degree at least 2 and no induced 6‐cycle, then γ_t(G) ≤ 6n/11. Both bounds are shown to be sharp. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 55–79, 2009     

Matching properties in connected domination critical graphs

A dominating set of vertices S of a graph G is connected if the subgraph G[S] is connected. Let γ_c(G) denote the size of any smallest connected dominating set in G. A graph G is k-γ-connected-critical if γ_c(G)=k, but if any edge is added to G, then γ_c(G+e)?k-1. This is a variation on the earlier concept of criticality of edge addition with respect to ordinary domination where a graph G was defined to be k-critical if the domination number of G is k, but if any edge is added to G, the domination number falls to k-1.A graph G is factor-critical if G-v has a perfect matching for every vertex v∈V(G), bicritical if G-u-v has a perfect matching for every pair of distinct vertices u,v∈V(G) or, more generally, k-factor-critical if, for every set S⊆V(G) with |S|=k, the graph G-S contains a perfect matching. In two previous papers [N. Ananchuen, M.D. Plummer, Matching properties in domination critical graphs, Discrete Math. 277 (2004) 1-13; N. Ananchuen, M.D. Plummer, 3-factor-criticality in domination critical graphs, Discrete Math. 2007, to appear [3].] on ordinary (i.e., not necessarily connected) domination, the first and third authors showed that under certain assumptions regarding connectivity and minimum degree, a critical graph G with (ordinary) domination number 3 will be factor-critical (if |V(G)| is odd), bicritical (if |V(G)| is even) or 3-factor-critical (again if |V(G)| is odd). Analogous theorems for connected domination are presented here. Although domination and connected domination are similar in some ways, we will point out some interesting differences between our new results for the case of connected domination and the results in [N. Ananchuen, M.D. Plummer, Matching properties in domination critical graphs, Discrete Math. 277 (2004) 1-13; N. Ananchuen, M.D. Plummer, 3-factor-criticality in domination critical graphs, Discrete Math. 2007, to appear [3].].     

Convex domination in the composition and cartesian product of graphs

In this paper we characterize the convex dominating sets in the composition and Cartesian product of two connected graphs. The concepts of clique dominating set and clique domination number of a graph are defined. It is shown that the convex domination number of a composition G[H] of two non-complete connected graphs G and H is equal to the clique domination number of G. The convex domination number of the Cartesian product of two connected graphs is related to the convex domination numbers of the graphs involved.     

Some Graphs with Double Domination Subdivision Number Three

A subset ${S \subseteq V(G)}$ is a double dominating set of G if S dominates every vertex of G at least twice. The double domination number dd(G) is the minimum cardinality of a double dominating set of G. The double domination subdivision number sd _dd (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 double domination number. Atapour et al. (Discret Appl Math, 155:1700–1707, 2007) posed an open problem: Prove or disprove: let G be a connected graph with no isolated vertices, then 1 ≤ sd _dd (G) ≤ 2. In this paper, we disprove the problem by constructing some connected graphs with no isolated vertices and double domination subdivision number three.     

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.     

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.     

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.     

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.     

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 bondage number of middle graphs

Let G = (V (G),E(G)) be a simple graph. A subset S of V (G) is a dominating set of G if, for any vertex v ∈ V (G) — S, there exists some vertex u ∈ S such that uv ∈ E(G). The domination number, denoted by γ(G), is the cardinality of a minimal dominating set of G. There are several types of domination parameters depending upon the nature of domination and the nature of dominating set. These parameters are bondage, reinforcement, strong-weak domination, strong-weak bondage numbers. In this paper, we first investigate the strong-weak domination number of middle graphs of a graph. Then several results for the bondage, strong-weak bondage number of middle graphs are obtained.