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.

In occasione della commemorazione dell’800-esimo anniversario della pubblicazione del Liber Abaci, desidero richiamare l’attenzione del lettore su alcuni dei fatti che preferisco riguardanti numeri di Fibonacci. Tali fatti includono la presenza di quadrati, di multipli di quadrati e di numeri potenti tra i numeri di Fibonacci, la rappresentazione di numeri reali e la costruzione di numeri trascendenti mediante numeri di Fibonacci, la possibilità di costruire una serie zeta ed un dominio a fattorizzazione...

In spectral bisection, a Fielder vector is used for partitioning a graph into two connected subgraphs according to its sign pattern. We investigate graphs having Fiedler vectors with unbalanced sign patterns such that a partition can result in two connected subgraphs that are distinctly different in size. We present a characterization of graphs having a Fiedler vector with exactly one negative component, and discuss some classes of such graphs. We also establish an analogous result for regular graphs...

We effectively construct in the Hilbert cube $ℍ={[0,1]}^{\omega }$ two sets $V,W\subset ℍ$ with the following properties: (a) $V\cap W=\varnothing$, (b) $V\cup W$ is discrete-dense, i.e. dense in ${{[0,1]}_D}^{\omega }$, where ${[0,1]}_D$ denotes the unit interval equipped with the discrete topology, (c) $V$, $W$ are open in $ℍ$. In fact, $V={\bigcup }_{\mathbb{N}}{V}_i$, $W={\bigcup }_{\mathbb{N}}{W}_i$, where $V_i={\bigcup }_0^{2^{i-1}-1}{V}_{ij}$, $W_i={\bigcup }_0^{2^{i-1}-1}{W}_{ij}$. $V_{ij}$, $W_{ij}$ are basic open sets and $(0,0,0,...)\in V_{ij}$, $(1,1,1,...)\in W_{ij}$, (d) $V_i\cup W_i$, $i\in \mathbb{N}$ is point symmetric about $(1/2,1/2,1/2,...)$. Instead of $[0,1]$ we could have taken any $T_4$-space or a digital interval, where the resolution (number of points) increases with $i$.

We study the concept of an $H$-partition of the vertex set of a graph $G$, which includes all vertex partitioning problems into four parts which we require to be nonempty with only external constraints according to the structure of a model graph $H$, with the exception of two cases, one that has already been classified as polynomial, and the other one remains unclassified. In the context of more general vertex-partition problems, the problems addressed in this paper have these properties: non-list, $4$-part,...

We study the concept of an H-partition of the vertex set of a graph G, which includes all vertex partitioning problems into four parts which we require to be nonempty with only external constraints according to the structure of a model graph H, with the exception of two cases, one that has already been classified as polynomial, and the other one remains unclassified. In the context of more general vertex-partition problems, the problems addressed in this paper have these properties: non-list, 4-part, external...