Domination graphs of regular tournaments

Domination Graphs of Tournaments and Digraphs

The domination graph of a digraph has the same vertices as the digraph with an edge between two vertices if every other vertex loses to at least one of the two. Previously, the authors showed that the domination graph of a tournament is either an odd cycle with or without isolated and/or pendant vertices, or a forest of caterpillars. They also showed that any graph consisting of an odd cycle wi...

Connected domination of regular graphs

A dominating set D 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 set C 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 set W of a g...

Minus domination in regular graphs

A three-valued function f defined on the vertices of a graph G = (V,E), f : V , ( 1 , 0 , 1), is a minus dominating function if the sum of its function values over any closed neighborhood is at least one. That is, for every v E V, f(N[v])>~ 1, where N[v] consists of v and every vertex adjacent to v. The weight of a minus dominating function is f ( V ) = ~ f (v) , over all vertices v E V. The mi...

Total domination of random regular graphs

A Characterization of Connected (1, 2)-Domination Graphs of Tournaments

Recently, Hedetniemi et al. introduced (1, 2)-domination in graphs, and the authors extended that concept to (1, 2)-domination graphs of digraphs. Given vertices x and y in a digraph D, x and y form a (1, 2)-dominating pair if and only if for every other vertex z in D, z is one step away from x or y and at most two steps away from the other. The (1, 2)-dominating graph of D, dom1,2 (D) , is def...