A set of vertices S is a determining set for a graph G if every automorphism of G is uniquely determined by its action on S. The determining number of a graph is the size of a smallest determining set. This paper describes ways of finding and verifying determining sets, gives natural lower bounds on the determining number, and shows how to use orbits to investigate determining sets. Further, de...

In this paper, a distributed model predictive control scheme is proposed for linear, time-invariant dynamically coupled systems. Uniquely, controllers optimize state and input constraint sets, and exchange information about these—rather than planned state and control trajectories—in order to coordinate actions and reduce the effects of the mutual disturbances induced via dynamic coupling. Mutua...

The reconstruction problem is considered in those classes of discrete sets where the reconstruction can be performed from two projections in polynomial time. The reconstruction algorithms and complexity results are summarized in the case of hv-convex sets, hv-convex 8-connected sets, hv-convex polyominoes, and directed h-convex sets. As new results some properties of the feet and spines of the ...

A set $Wsubset V (G)$ is called a resolving set, if for every two distinct vertices $u, v in V (G)$ there exists $win W$ such that $d(u,w) not = d(v,w)$, where $d(x, y)$ is the distance between the vertices $x$ and $y$. A resolving set for $G$ with minimum cardinality is called a metric basis. A graph with a unique metric basis is called a uniquely dimensional graph. In this paper, we establish...

A mixed hypergraph consists of two families of edges: the C-edges and D-edges. In a coloring every C-edge has at least two vertices of the same color, while every D-edge has at least two vertices colored differently. The largest and smallest possible numbers of colors in a coloring are termed the upper and lower chromatic number, χ̄ and χ, respectively. A mixed hypergraph is called uniquely colo...

We consider the Schrödinger operator on a finite interval with an $L^1$-potential. prove that potential can be uniquely recovered from one spectrum and subsets of another point masses spectral measure (or norming constants) corresponding to first spectrum. also solve this Borg–Marchenko-type problem under some conditions two spectra, when missing part second known have different index sets.