Domination dot-critical graphs

A graph G is dot-critical if contracting any edge decreases the domination number. It is totally dot-critical if identifying any two vertices decreases the domination number. We show that the totally dot-critical graphs essentially include the much-studied domination vertex-critical and edge-critical graphs as special cases. We investigate these properties, and provide a characterization of dot-critical and totally dot-critical graphs with domination number 2. We also consider the question of when a dot-critical graph contains a critical vertex.     

Upper Bound on the Diameter of a Domination Dot-Critical Graph

A vertex of a graph is called critical if its deletion decreases the domination number, and an edge is called dot-critical if its contraction decreases the domination number. A graph is said to be dot-critical if all of its edges are dot-critical. In this paper, we show that if G is a connected dot-critical graph with domination number k??? 3 and diameter d and if G has no critical vertices, then d??? 2k?3.     

Domination dot-critical graphs with no critical vertices

A graph G is dot-critical if contracting any edge decreases the domination number. It is totally dot-critical if identifying any two vertices decreases the domination number. Burton and Sumner [Discrete Math. 306 (2006) 11-18] posed the problem: Is it true that for k?4, there exists a totally k-dot-critical graph with no critical vertices? In this paper, we show that this problem has a positive answer.     

A note on the domination dot-critical graphs

A graph G is dot-critical if contracting any edge decreases the domination number. Nader Jafari Rad (2009) [3] posed the problem: Is it true that a connected k-dot-critical graph G with G^′=0? is 2-connected? In this note, we give a family of 1-connected 2k-dot-critical graph with G^′=0? and show that this problem has a negative answer.     

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.     

Codiameters of 3‐connected 3‐domination critical graphs

A graph G is 3‐domination critical if its domination number γ is 3 and the addition of any edge decreases γ by 1. Let G be a 3‐connected 3‐domination critical graph of order n. In this paper, we show that there is a path of length at least n?2 between any two distinct vertices in G and the lower bound is sharp. © 2002 John Wiley & Sons, Inc. J Graph Theory 39: 76–85, 2002     

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.     

On matching and total domination in graphs

A set M of edges of a graph G is a matching if no two edges in M are incident to the same vertex. A set S of vertices in G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The matching number is the maximum cardinality of a matching of G, while the total domination number of G is the minimum cardinality of a total dominating set of G. In this paper, we investigate the relationships between the matching and total domination number of a graph. We observe that the total domination number of every claw-free graph with minimum degree at least three is bounded above by its matching number, and we show that every k-regular graph with k?3 has total domination number at most its matching number. In general, we show that no minimum degree is sufficient to guarantee that the matching number and total domination number are comparable.     

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].].     

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.     

Independence and connectivity in 3-domination-critical graphs

Let δ, γ, κ and α be, respectively, the minimum degree, the domination number, the connectivity and the independence number of a graph G. The graph G is 3-domination-critical if γ=3 and the addition of any edge decreases γ by 1. In this paper, we prove that if G is a 3-domination-critical graph, then α⩽κ+2; and moreover, if κ⩽δ−1, then α⩽κ+1. We also give a short proof of Wojcicka's result, which says that every connected 3-domination-critical graph of order at least 7 contains a hamiltonian path (J. Graph Theory 14 (1990) 205).     

Perfect Matchings in Total Domination Critical Graphs

A graph is total domination edge-critical if the addition of any edge decreases the total domination number, while a graph with minimum degree at least two is total domination vertex-critical if the removal of any vertex decreases the total domination number. A 3 _t EC graph is a total domination edge-critical graph with total domination number 3 and a 3 _t VC graph is a total domination vertex-critical graph with total domination number 3. A graph G is factor-critical if G − v has a perfect matching for every vertex v in G. In this paper, we show that every 3 _t EC graph of even order has a perfect matching, while every 3 _t EC graph of odd order with no cut-vertex is factor-critical. We also show that every 3 _t VC graph of even order that is K _1,7-free has a perfect matching, while every 3 _t VC graph of odd order that is K _1,6-free is factor-critical. We show that these results are tight in the sense that there exist 3 _t VC graphs of even order with no perfect matching that are K _1,8-free and 3 _t VC graphs of odd order that are K _1,7-free but not factor-critical.     

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).     

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.     

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.     

Domination Defect in Graphs: Guarding With Fewer Guards

In this paper, we introduce a new graph parameter called the domination defect of a graph. The domination number γ of a graph G is the minimum number of vertices required to dominate the vertices of G. Due to the minimality of γ, if a set of vertices of G has cardinality less than γ then there are vertices of G that are not dominated by that set. The k-domination defect of G is the minimum number of vertices which are left un-dominated by a subset of γ - k vertices of G. We study different bounds on the k-domination defect of a graph G with respect to the domination number, order, degree sequence, graph homomorphisms and the existence of efficient dominating sets. We also characterize the graphs whose domination defect is 1 and find exact values of the domination defect for some particular classes of 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.     

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.