An edge-colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. It was proved that computing rc(G) is an NP-Hard problem, as well as that even deciding whether a graph has rc(G) =...

Let G be a graph with the vertex set V (G) and edge set E(G). A function f : E(G) → {−1,+1} is said to be a signed star dominating function ofG if ∑ e∈EG(v) f(e) ≥ 1, for every v ∈ V (G), where EG(v) = {uv ∈ E(G) |u ∈ V (G)}. The minimum of the values of ∑ e∈E(G) f(e), taken over all signed star dominating functions f on G is called the signed star domination number of G and is denoted by γss(G...

Let k be a positive integer and G = (V,E) be a connected graph of order n. A set D ⊆ V is called a k-dominating set of G if each x ∈ V (G) − D is within distance k from some vertex of D. A connected k-dominating set is a k-dominating set that induces a connected subgraph of G. The connected k-domination number of G, denoted by γ k(G), is the minimum cardinality of a connected k-dominating set. ...

An exponential dominating set of graph G = (V,E) is a subset D ⊆ V such that ∑ w∈D( 1 2 )d(v,w)−1 ≥ 1 for every v ∈ V, where d(v, w) is the distance between vertices v and w. The exponential domination number, γe(G), is the smallest cardinality of an exponential dominating set. Lower and upper bounds for γe(Cm × Cn) are determined and it is shown that limm,n→∞ γe(Cm×Cn) mn ≤ 1 13 . Two connecti...

For graphs G and H, let G?rbH denote the property that, for every proper edge-colouring of G, there is a rainbow H in G. graph threshold function pHrb=pHrb(n) this random G(n,p) satisfies pHrb=O(n?1/m(2)(H)), where m(2)(H) denotes so-called maximum 2-density H. Completing result Nenadov, Person, Škori?, Steger [J. Combin. Theory Ser. B 124 (2017), 1–38], we prove matching lower bound pKkrb k?5....

A function f : V (G) → {+1,−1} defined on the vertices of a graph G is a signed dominating function if for any vertex v the sum of function values over its closed neighborhood is at least 1. The signed domination number γs(G) of G is the minimum weight of a signed dominating function on G. By simply changing “{+1,−1}” in the above definition to “{+1, 0,−1}”, we can define the minus dominating f...

Electro-olfactogram recording was used to determine whether the olfactory epithelium of adult rainbow trout is specifically sensitive to bile acids, some of which have been hypothesized to function as pheromones. Of 38 bile acids that had been pre-screened for olfactory activity, 6 were selected. The rainbow trout-specific bile acids, taurocholic acid (TCA), and taurolithocholic acid 3-sulfate ...

Given a graph, a connected dominating set is a subset of vertices such that every vertex is either in the subset or adjacent to a vertex in the subset and the subgraph induced by the subset is connected.A minimum connected dominating set is such a vertex subset with minimum cardinality. In this paper, we present a new one-step greedy approximation with performance ratio ln + 2 where is the maxi...

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...

Cryptanalytic time memory tradeoff is a tool for inverting one-way functions, and the rainbow table method, the best-known tradeoff algorithm, is widely used to recover passwords. Even though extensive research has been performed on the rainbow tradeoff, the algorithm actually used in practice differs from the well-studied original algorithm. This work provides a full analysis of the rainbow tr...