Michael A. Henning

A survey of selected recent results on total domination in 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. In this paper, we offer a survey of selected recent results on total domination in graphs.

Domination in Graphs Applied to Electric Power Networks

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. We consider the graph theoretical representation of this problem as a variation of the dominating set problem and define 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 minimum cardinality of a power dominating set of a graph G is the power domination number

Graphs with large total domination number

\gamma_P(G)

Locating and total dominating sets in trees

. We show that the power dominating set (PDS) problem is NP-complete even when restricted to bipartite graphs or chordal graphs. On the other hand, we give a linear algorithm to solve the PDS for trees. In addition, we investigate theoretical properties of

Paired-Domination in Claw-Free Cubic Graphs

\gamma_P(T)

Total domination in graphs with minimum degree three

in trees T.

A characterization of roman trees

Let G = (V,E) be a graph. A set S ⊆ V is a total dominating set if every vertex of V is adjacent to some vertex in S. The total domination number of G, denoted by Υt(G), is the minimum cardinality of a total dominating set of G. We establish a property of minimum total dominating sets in graphs. If G is a connected graph of order n ≥ 3, then (see [3]) Υt(G) ≤ 2n-3. We show that if G is a connected graph of order n with minimum degree at least 2, then either Υt(G) ≤ 4n-7 or G e {C3, C5, C6, C10}. A characterization of those graphs of order n which are edge-minimal with respect to satisfying G connected, δ(G) e 2 and Υt(G) ≥ 4n-7 is obtained. We establish that if G is a connected graph of size q with minimum degree at least 2, then Υt(G) ≤(q + 2)-2. Connected graphs G of size q with minimum degree at least 2 satisfying Υt(G) > q-2 are characterized.

The diameter of total domination vertex critical graphs

A set S of vertices in a graph G = (V, E) is a total dominating set of G if every vertex of V is adjacent to a vertex in S. We consider total dominating sets of minimum cardinality which have the additional property that distinct vertices of V are totally dominated by distinct subsets of the total dominating set.

k-tuple total domination in graphs

Abstract.A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The minimum cardinality of a paired-dominating set of G is the paired-domination number of G, denoted by γpr(G). If G does not contain a graph F as an induced subgraph, then G is said to be F-free. In particular if F=K1,3 or K4−e, then we say that G is claw-free or diamond-free, respectively. Let G be a connected cubic graph of order n. We show that (i) if G is (K1,3,K4−e,C4)-free, then γpr(G)≤3n/8; (ii) if G is claw-free and diamond-free, then γpr(G)≤2n/5; (iii) if G is claw-free, then γpr(G)≤n/2. In all three cases, the extremal graphs are characterized.

A note on power domination in grid graphs

We consider the equivalence classes of graphs induced by the unsigned versions of the Reidemeister moves on knot diagrams. Any graph that is reducible by some finite sequence of these moves, to a graph with no edges, is called a knot graph. We show that the class of knot graphs strictly contains the set of delta-wye graphs. We prove that the dimension of the intersection of the cycle and cocycle spaces is an effective numerical invariant of these classes.