For various domination-related parameters involving locating devices (distinguishing sets) that function as places from which detectors can determine information about the location of an “intruder”, several types of possible detector faults are identified. Two of these fault tolerant detector types for distinguishing sets are considered here, namely redundant distinguishing and detection distinguishing. Illustrating these concepts, we focus primarily on open-locating-dominating sets.

Let G = (V,E) be a connected graph and let k be a positive integer with k ≤ rad(G). A subset D ⊆ V is called a distance k-dominating set of G if for every v ∈ V - D, there exists a vertex u ∈ D such that d(u,v) ≤ k. In this paper we study the fractional version of distance k-domination and related parameters.

Mynhardt has conjectured that if G is a graph such that γ(G) = γ(πG) for all generalized prisms πG then G is edgeless. The fractional analogue of this conjecture is established and proved by showing that, if G is a graph with edges, then ${\gamma }_f\left(G\times K₂\right)>{\gamma }_f\left(G\right)$.

Let G = (V,E) be a graph. A function g:V → [0,1] is called a global dominating function (GDF) of G, if for every v ∈ V, $g\left(N\left[v\right]\right)={\sum }_{u\in N\left[v\right]}g\left(u\right)\ge 1$ and $g\left(\overline{N\left(v\right)}\right)={\sum }_{u\notin N\left(v\right)}g\left(u\right)\ge 1$. A GDF g of a graph G is called minimal (MGDF) if for all functions f:V → [0,1] such that f ≤ g and f(v) ≠ g(v) for at least one v ∈ V, f is not a GDF. The fractional global domination number ${\gamma }_{fg}\left(G\right)$ is defined as follows: ${\gamma }_{fg}\left(G\right)$ = min|g|:g is an MGDF of G where $\left|g\right|={\sum }_{v\in V}g\left(v\right)$. In this paper we initiate a study of this parameter.

For each vertex v in a graph G, let there be associated a subgraph $H_v$ of G. The vertex v is said to dominate $H_v$ as well as dominate each vertex and edge of $H_v$. A set S of vertices of G is called a full dominating set if every vertex of G is dominated by some vertex of S, as is every edge of G. The minimum cardinality of a full dominating set of G is its full domination number ${\gamma }_{FH}\left(G\right)$. A full dominating set of G of cardinality ${\gamma }_{FH}\left(G\right)$ is called a ${\gamma }_{FH}$-set of G. We study three types of full domination in graphs: full...