A total dominating set of a graph G is a set D of vertices of G such that every vertex of G has a neighbor in D. A locating-total dominating set of G is a total dominating set D of G with the additional property that every two distinct vertices outside D have distinct neighbors in D; that is, for distinct vertices u and v outside D, N(u) ∩D 6= N(v) ∩D where N(u) denotes the open neighborhood of...

A mixed dominating set is a collection of vertices and edges that dominates all graph. We study the complexity exact parameterized algorithms for \textsc{Mixed Dominating Set}, resolving some open questions. In particular, we settle problem's by treewidth pathwidth giving an algorithm running in time $O^*(5^{tw})$ (improving current best $O^*(6^{tw})$), as well lower bound showing our cannot be...

Let G be a graph. A total dominating set of G is a set S of vertices of G such that every vertex is adjacent to at least one vertex in S. The total domatic number of a graph is the maximum number of total dominating sets which partition the vertex set of G. In this paper we would like to characterize the cubic graphs with total domatic number at least two.

A subset X of edges in a graph G is called an edge dominating set of G if every edge not in X is adjacent to some edge in X. The edge domination number γ′(G) of G is the minimum cardinality taken over all edge dominating sets of G. Let m,n and k be positive integers with n − 1 ≤ m ≤ (n 2 ) , G(m,n) be the set of all non-isomorphic connected graphs of order n and size m, and G(m,n; k) = {G ∈ G(m...

Let G = (V,E) be a graph. A set D ⊆ V is a total outer-connected dominating set of G if D is dominating and G[V −D] is connected. The total outer-connected domination number of G, denoted γtc(G), is the smallest cardinality of a total outer-connected dominating set of G. It is known that if T is a tree of order n ≥ 2, then γtc(T ) ≥ 2n 3 . We will provide a constructive characterization for tre...