Irredundance has been studied in the context of dominating sets, via the concept of a private neighbor. Here irredundance of zero forcing sets is introduced via the concept of a private fort and the upper and lower zero forcing irrdedundance numbers ZIR(G) and zir(G) are defined. Bounds on ZIR(G) and zir(G) are established and graphs having extreme values of ZIR(G) and zir(G) are characterized. The effect of the join and corona operations is studied. As the concept of a zero forcing irrdedundant set is new, there are many questions for future research.

Let $G$ be a graph and c a proper k-coloring of G, i.e. any two adjacent vertices u and v have different colors c(u) and c(v). A proper k-coloring is a b-coloring if there exists a vertex in every color class that contains all the colors in its closed neighborhood. The maximum number of colors k admitting b-coloring of G is the b-chromatic number. We present two separate approaches to the conjecture posed by Blidia et. al that the b-chromatic number equals to d+1 for every d-regular graph of girth at least five except the Petersen graph.

A subset $D$ of $V$ is \emph{dominating} in $G$ if every vertex of $V-D$ has at least one neighbour in $D;$ let $\gamma(G)$ be the minimum cardinality among all dominating sets in $G.$ A graph $G$ is $\gamma$-$q$-{\it critical} if the smallest subset of edges whose subdivision necessarily increases $\gamma(G)$ has cardinality $q.$ In this paper we consider mainly $\gamma$-$q$-critical trees and give some general properties of $gamma$-$q$-critical graphs. In particular, we show that if $T$ is a $\gamma$-$q$-critical tree, then $1 \leq q \leq n(T)-1$ and we characterize extremal trees when $q=n(T)-1.$ Since a subdivision number {of a tree $T$} ${\rm sd}(T)$ is always $1,2$ or $3,$ we also characterize $\gamma$-2-critical trees $T$ with ${\rm sd}(T)=2$ and $\gamma$-3-critical trees $T$ with ${\rm sd}(T)=3.$

A set D of vertices of a graph G with vertex set V is irredundant if each non-isolated vertex of G[D] has a neighbour in V-D that is not adjacent to any other vertex in D. The upper irredundance number IR(G) is the largest cardinality of an irredundant set of G; an IR(G)-set is an irredundant set of cardinality IR(G). The IR-graph of G has the IR(G)-sets as vertex set, and sets A and B are adjacent if and only if B can be obtained from A by exchanging a single vertex of A for an adjacent vertex in B. An IR-tree is an IR-graph that is a tree. We characterize IR-trees of diameter 3 by showing that these graphs are precisely the double stars S(2n,2n), i.e., trees obtained by joining the central vertices of two disjoint stars K_1,2n.

A set D of vertices of a graph G = ( V , E ) is irredundant if each v ∈ D satisfies (a) v is isolated in the subgraph induced by D, or (b) v is adjacent to a vertex in V − D that is nonadjacent to all other vertices in D. The upper irredundance number IR ( G ) is the largest cardinality of an irredundant set of G; an IR ( G )-set is an irredundant set of cardinality IR ( G ). The IR-graph of G has the irredundant sets of G of maximum cardinality, that is, the IR ( G )-sets, as vertex set, and sets D and D ′ are adjacent if and only if D ′ is obtained from D by exchanging a single vertex of D for an adjacent vertex in D ′. We study the realizability of graphs as IR-graphs and show that all disconnected graphs are IR-graphs, but some connected graphs (e.g. stars K 1 , n , n ≥ 2, P 4 , P 5 , C 5 , C 6 , C 7) are not. We show that the double star S ( 2 , 2 ) – the tree obtained by joining the two central vertices of two disjoint copies of P 3 – is the unique smallest IR-tree with diameter 3 and also a smallest non-complete IR-tree, and the tree obtained by subdividing a single pendant edge of S ( 2 , 2 ) is the unique smallest IR-tree with diameter 4.

The localization game is played by two players: a Cop with a team of k cops, and a Robber. The game is initialized by the Robber choosing a vertex r∈V, unknown to the Cop. Thereafter, the game proceeds turn based. At the start of each turn, the Cop probes k vertices and in return receives a distance vector. If the Cop can determine the exact location of r from the vector, the Robber is located and the Cop wins. Otherwise, the Robber is allowed to either stay at r, or move to r′ in the neighbourhood of r. The Cop then again probes k vertices. The game continues in this fashion, where the Cop wins if the Robber can be located in a finite number of turns. The localization number ζ(G), is defined as the least positive integer k for which the Cop has a winning strategy irrespective of the moves of the Robber. In this paper, we focus on the game played on Cartesian products. We prove that ζ(G□H)≥max{ζ(G),ζ(H)} as well as ζ(G□H)≤ζ(G)+ψ(H)−1, where ψ(H) is the doubly resolving number of H. We also show that ζ(Cm□Cn) is mostly equal to two.

For a graph G = (V, E), the k-dominating graph D-k(G) of G has vertices corresponding to the dominating sets of G having cardinality at most k, where two vertices of D-k(G) are adjacent if and only if the dominating set corresponding to one of the vertices can be obtained from the dominating set corresponding to the second vertex by the addition or deletion of a single vertex. We denote the domination and upper domination numbers of G by gamma(G) and Gamma(G), respectively, and the smallest integer epsilon for which D-k(G) is connected for all k >= epsilon by d(0)(G). It is known that Gamma(G) + 1 <= d(0)(G) <= vertical bar V vertical bar, but constructing a graph G such that d(0)(G) > Gamma(G) + 1 appears to be difficult. We present two related constructions. The first construction shows that for each integer k >= 3 and each integer r such that 1 <= r <= k - 1, there exists a graph G(k,r) such that Gamma(G(k,r)) = k, gamma(G(k,r)) = r+1 and d(0)(G(k,r)) = k+r = Gamma(G)+gamma(G)-1. The second construction shows that for each integer k >= 3 and each integer r such that 1 <= r <= k - 1, there exists a graph Q(k,r) such that Gamma(Q(k,r)) = k,gamma(Q(k,r)) = r and d(0)(Q(k,r)) = k + r = Gamma(G)+gamma(G). (C) 2018 Elsevier B.V. All rights reserved.

A broadcast on a nontrivial connected graph G=(V,E) is a function f from V(G) to {0,1,...,diam(G)} such that f(v) does not exceed the eccentricity of v. The cost of f is the sum of the function values. A broadcast f is dominating if each vertex of G is at distance at most f(v) from a vertex v with positive f(v). We use properties of minimal dominating broadcasts to define the concept of an irredundant broadcast on G. We determine conditions under which an irredundant broadcast is maximal irredundant. Denoting the minimum costs of dominating and maximal irredundant broadcasts by gamma_{b}(G) and ir_{b}(G) respectively, the definitions imply that ir_{b}(G) is bounded above by gamma_{b}(G) for all graphs. We show that gamma_{b} in turn is bounded above by (5/4)ir_{b}(G) for all graphs G. We also briefly consider the upper broadcast number Gamma_{b}(G) and upper irredundant broadcast number IR_{b}(G), and illustrate that the ratio IR_{b} to Gamma_{b} is unbounded for general graphs.

Let G = (V, E) be a simple graph of order n with vertex set V = {v(1), . . . , v(n)} and suppose that at most r(i) units of some commodity may be placed at any vertex v(i) while at least s(i) units must be placed in the vicinity (i.e. closed neighbourhood) of v(i) for i = 1, . . . , n. The smallest number of units that may be placed on the vertices of the graph satisfying the above requirements is called the < r, s >-domination number of the graph. In this paper we present a linear-time algorithm, using linear space, for determining the < r, s >-domination number of any tree.

Let G=(V,E) be a simple graph of order n with vertex set V={v1,…,vn} and suppose that at most ri units of some commodity may be placed at any vertex vi while at least si units must be placed in the closed neighbourhood of vi for i=1,…,n. The smallest number of units that may be placed on the vertices of the graph satisfying the above requirements is called the 〈r,s〉-domination number of the graph. The case where r=[r,…,r] and s=[s,…,s] is called the balanced case of 〈r,s〉-domination. We establish three upper bounds on the 〈r,s〉-domination number of a graph for the balanced case in this paper.

The domination number γ(G) of a graph G is the least number of vertices in a dominating set of G, and the lower irredundance number ir(G) is the least number of vertices in a maximal irredundant set of G. For each of these graph parameters, we establish bounds on the parameter of the coalescence of two graphs in terms of the parameters of the two respective graphs. These results are then utilised in the construction of graphs that are γ-critical but not ir-critical. Such graphs were not previously known to exist.

The two ARTEMIS probes observe significant precursor activity upstream from the Moon, when magnetically connected to the dayside lunar surface. The most common signature consists of high levels of whistler wave activity near half of the electron cyclotron frequency. This precursor activity extends to distances of many thousands of km, in both the solar wind and terrestrial magnetosphere. In the magnetosphere, electrons reflect from a combination of magnetic and electrostatic fields above the lunar surface, forming loss cone distributions. In the solar wind they generally form conics, as a result of reflection from an obstacle moving with respect to the plasma frame (just as at a shock). The anisotropy associated with these reflected electrons provides the free energy source for the whistlers, with cyclotron resonance conditions met between the reflected source population and Moonward‐propagating waves. These waves can in turn affect incoming plasma, and we observe significant perpendicular electron heating and plasma density depletions in some cases. In the magnetosphere, we also observe broadband electrostatic modes driven by beams of secondary electrons and/or photoelectrons accelerated outward from the surface. We also occasionally see waves near the ion cyclotron frequency in the magnetosphere. These lower frequency waves, which may result from the presence of ions of lunar origin, modulate the whistlers described above, as well as the electrons. Taken together, our observations suggest that the presence of the Moon leads to the formation of an upstream region analogous in many ways to the terrestrial electron foreshock.

Taking advantage of the string-of-pearls configuration of the five THEMIS spacecraft during the early phase of their mission, we analyze observations taken simultaneously in the magnetosheath, the magnetopause current layer and the magnetosphere. We find that electron heating coincides with ultra low frequency waves. It seems unlikely that electrons are heated by these waves because the electron thermal velocity is much larger than the Alfvén velocity (Va). In the short transverse scale (k&bot;ρi >> 1) regime, however, short scale Alfvén waves (SSAWs) have parallel phase velocities much larger than Va and are shown to interact, via Landau damping, with electrons thereby heating them. The origin of these waves is also addressed. THEMIS data give evidence for sharp spatial gradients in the magnetopause current layer where the highest amplitude waves have a large component δB perpendicular to the magnetopause and k azimuthal. We suggest that SSAWs are drift waves generated by temperature gradients in a high beta, large Ti/Te magnetopause current layer. Therefore these waves are called SSDAWs, where D stands for drift. SSDAWs have large k&bot; and therefore a large Doppler shift that can exceed their frequencies in the plasma frame. Because they have a small but finite parallel electric field and a magnetic component perpendicular to the magnetopause, they could play a key role at reconnecting magnetic field lines. The growth rate depends strongly on the scale of the gradients; it becomes very large when the scale of the electron temperature gradient gets below 400 km. Therefore SSDAW's are expected to limit the sharpness of the gradients, which might explain why Berchem and Russell (1982) found that the average magnetopause current sheet thickness to be ~400–1000 km (~500 km in the near equatorial region).

Electron phase-space holes (EHs) are indicators of nonlinear activities in space plasmas. Most often they are observed as electrostatic signals, but recently Andersson et al. [2009] reported electromagnetic EHs observed by the THEMIS mission in the Earth's plasma sheet. As a follow-up to Andersson et al. [2009], this paper presents a model of electromagnetic EHs where the delta E x B-0 drift of electrons creates a net current. The model is examined with test-particle simulations and compared to the electromagnetic EHs reported by Andersson et al. [2009]. As an application of the model, we introduce a more accurate method than the simplified Lorentz transformation of Andersson et al. [2009] to derive EH velocity (v(EH)). The sizes and potentials of EHs are derived from v(EH), so an accurate derivation of v(EH) is important in analyzing EHs. In general, our results are qualitatively consistent with those of Andersson et al. [2009] but generally with smaller velocities and sizes.