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The total domination number of a graph is the size of a smallest total dominating set, where a total dominating set is a set of vertices of the graph such that all vertices (including those in the set itself) have a neighbor in the set. Total dominating numbers are defined only for graphs having no isolated vertex (plus the trivial case of the singleton graph ).For example, in the Petersen graph illustrated above, since the set is a minimum dominating set (left figure), while since is a minimum total dominating set (right figure).For any simple graph with no isolated points, the total domination number and ordinary domination number satisfy(1)(Henning and Yeo 2013, p. 17). In addition, if is a bipartite graph, then(2)(Azarija et al. 2017), where denotes the graph Cartesian product.For a connected graph with vertex count ,(3)(Cockayne et al. 1980, Henning and Yeo 2013, p. 11)...

For a graph and a subset of the vertex set , denote by the set of vertices in which are adjacent to a vertex in . If , then is said to be a total dominating set (of vertices in ). Because members of a total dominating set must be adjacent to another vertex, total dominating sets are not defined for graphs having an isolated vertex.The total dominating set differs from the ordinary dominating set in that in a total dominating set , the members of are required to themselves be adjacent to a vertex in , whereas is an ordinary dominating set , the members of may be either in itself or adjacent to vertices in .For example, in the Petersen graph illustrated above, the set is a (minimum) dominating set (left figure), while is a (minimum) total dominating set (right figure).The size of a minimum total dominating set is called the total domination number...

Let be the number of dominating sets of size in a graph , then the domination polynomial of in the variable is defined aswhere is the domination number of (Kotek et al. 2012, Alikhani and Peng 2014). is multiplicative over connected components (Alikhani and Peng 2014).Precomputed dominations polynomials for many named graphs in terms of a variable and in the Wolfram Language as GraphData[graph, "DominationPolynomial"][x].The following table summarizes closed forms for the domination polynomials of some common classes of graphs (cf. Alikhani and Peng 2014).graphbarbell graphbook graph cocktail party graph complete bipartite graph complete graph empty graph helm graphsunlet graph The following table summarizes the recurrence relations for domination polynomials for some simple classes of graphs.graphorderrecurrenceantiprism graph5barbell graph3book graph 3cocktail party graph 3complete graph 2cycle graph 3empty..

The domination number of a graph , denoted , is the minimum size of a dominating set of vertices in , i.e., the size of a minimum dominating set. The domination number is also equal to smallest exponent in a domination polynomial. For example, in the Petersen graph illustrated above, the set is a minimum dominating set, so .The domination number should not be confused with the domatic number, which is the maximum size of a domatic partition in a graph.There are several variations of the domination number originating from variations of the underlying dominating set, the most prevalent being the total domination number (which is the minimum size of a total dominating set).The complete graphs (each vertex is adjacent to every other), star graphs (the central vertex is adjacent to all leaves), and the wheel graph (the central vertex is adjacent to all rim vertices) all have domination number 1 by construction.The domination number satisfies(1)where..

For a graph and a subset of the vertex set , denote by the set of vertices in which are in or adjacent to a vertex in . If , then is said to be a dominating set (of vertices in ).A dominating set of smallest size is called a minimum dominating set and its size is known as the domination number. A dominating set that is not a proper subset of any other dominating set is called a minimal dominating set.For example, in the Petersen graph illustrated above, the set is a dominating set (and, in fact, a minimum dominating set).The domination polynomial encodes the numbersof dominating sets of various sizes.Other variants of the usual dominating set can be defined, including the so-calledtotal dominating set.Precomputed dominating sets for many named graphs can be obtained in the Wolfram Language using GraphData[graph, "DominatingSets"]...

The maximum number of disjoint dominating sets in a domatic partition of a graph is called its domatic number .The domatic number should not be confused with the domination number, which is the size of the smallest individual dominating set.Let be the minimum vertex degree of a graph , then The domatic number of a graph with one or more isolatedpoints is therefore 1.Furthermore, if the domination number of a graph is known, thanwhere denotes the vertex count of and is the floor function.Finding the domatic number of a graph is computationally hard.Given a complete set of minimal dominating sets, the domatic number of a graph can be found as the independence number of the graph in which vertices are minimal dominating sets of and edges exist between pairs sets having nonempty intersection...

The connected domination number of a connected graph , denoted , is the size of a minimum connected dominating set of a graph .The maximum leaf number and connected domination number of a graph are connected bywhere is the vertex count of .Many families of graphs have simple closed forms, as summarized in the following table. In the table, denotes the floor function.graph familyconnected domination numberAndrásfai graphApollonian networkantiprism graphbarbell graph2black bishop graph book graph 2cocktail party graph 2complete bipartite graph complete bipartite graph 2complete graph 1complete tripartite graph complete tripartite graph 2-crossed prism graphcrown graph 4cycle graph gear graphhelm graphladder graph Möbius ladder pan graphpath graph prism graph rook complement graph rook graph star graph 1sun graphsunlet graph triangular graphweb graphwheel graph 1white bishop graph ..

A connected dominating set in a connected graph is a dominating set in whose vertices induce a connected subgraph, i.e., one in which there is no dominating vertex not connected to some other dominating vertex by an edge. Connected dominating sets therefore comprise a subset of all dominating sets in a graph.A minimum connected dominating set of a graph is a connected dominating set of smallest possible size, where the minimum size is denoted and known as the connected domination number.

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