A Roman dominating function (RDF) on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex v for which f(v) = 0, is adjacent to at least one vertex u for which f(u) = 2. The weight of a Roman dominating function f is the value f(V (G)) = ∑ v∈V (G) f(v). The Roman domination number of G, denoted by γR(G), is the minimum weight of an RDF on G. The Roman reinforc...

Given an undirected graph G = (V,E) and a weight function w : E → R, the Minimum Dominating Tree problem asks to find a minimum weight sub-tree of G, T = (U,F ), such that every v ∈ V \U is adjacent to at least one vertex in U . The special case when the weight function is uniform is known as the Minimum Connected Dominating Set problem. Given an undirected graph G = (V,E) with some subsets of ...

In this short note, we study pairwise edge-disjoint rainbow spanning trees in properly edge-coloured complete graphs, where a graph is rainbow if its edges have distinct colours. Brualdi and Hollingsworth conjectured that every Kn properly edge-coloured by n−1 colours has n/2 edge-disjoint rainbow spanning trees. Kaneko, Kano and Suzuki later suggested this should hold for every properly edge-c...

A signed Roman dominating function on the digraphD is a function f : V (D) −→ {−1, 1, 2} such that ∑ u∈N−[v] f(u) ≥ 1 for every v ∈ V (D), where N−[v] consists of v and all inner neighbors of v, and every vertex u ∈ V (D) for which f(u) = −1 has an inner neighbor v for which f(v) = 2. A set {f1, f2, . . . , fd} of distinct signed Roman dominating functions on D with the property that ∑d i=1 fi(...

The susceptibility of 2 strains of rainbow trout Oncorhynchus mykiss, 1 from North America (TL) and 1 from Germany (GR), to Myxobolus cerebralis (the cause of salmonid whirling disease) was assessed following exposure to the infectious stages (triactinomyxons). Two laboratory experiments were conducted with age-matched rainbow trout of each strain. At the beginning of the study, the 2 trout str...

To explore how the cardiac sarcoplasmic reticulum (SR) functions over a range of temperatures, we used whole-cell voltage clamp combined with rapid caffeine application to study SR Ca(2+) accumulation, release and steady-state content in atrial myocytes from rainbow trout. Myocytes were isolated from rainbow trout acclimated to 14 degrees C, and the effect of varying stimulation pulse number, f...

A two-valued function f defined on the vertices of a graph G = (V,E), f : V → {−1, 1}, is a signed total dominating function if the sum of its function values over any open neighborhood is at least one. That is, for every v ∈ V, f(N(v)) ≥ 1, where N(v) consists of every vertex adjacent to v. The weight of a total signed dominating function is f(V ) = ∑ f(v), over all vertices v ∈ V . The total ...

A path in an edge colored graph is said to be a rainbow path if no two edges on the path have the same color. An edge colored graph is (strongly) rainbow connected if there exists a (geodesic) rainbow path between every pair of vertices. The (strong) rainbow connectivity of a graph G, denoted by (src(G), respectively) rc(G) is the smallest number of colors required to edge color the graph such ...

An edge-coloured graph G is said to be rainbow-connected if any two vertices are connected by a path whose edges have different colours. The rainbow connection number of a graph is the minimum number of colours needed to make the graph rainbow-connected. This graph parameter was introduced by G. Chartrand, G.L. Johns, K.A. McKeon and P. Zhang in 2008. Since, the topic drew much attention, and v...