In the convergecast scheduling problem, it is required to find a spanning aggregation tree with a root in a base station (sink) in a given communication graph and build a conflict-free (half-duplex, interference-aware) min-length schedule for aggregating data over the edges of the aggregation tree. This problem is strongly NP-hard even in the case of a given aggregation tree. However, if the co...

The infinite grid is the Cayley graph of Z × Z with the usual generators. In this paper, the Ihara zeta function for the infinite grid is computed using elliptic integrals and theta functions. The zeta function of the grid extends to an analytic, multivalued function which satisfies a functional equation. The set of singularities in its domain is finite. The grid zeta function is the first comp...

There are several scenarios in which a given drawing of a graph is to be modified subject to preservation constraints. Examples include shape simplification, sketch-based, and dynamic graph layout. While the orthogonal ordering of vertices is a natural and frequently called for preservation constraint, we show that, unfortunately, it results in severe algorithmic difficulties even for the simpl...

Contact and intersection representations of graphs and particularly of planar graphs have been studied for decades. The by now best known result in the area may be the Koebe-Andreev-Thurston circle packing theorem. A more recent highlight in the area is a result of Chalopin and Gonçalves: every planar graph is an intersection graph of segments in the plane. This boosted the study of intersectio...

The Supervisory Control and Data Acquisition (SCADA) State Estimator (SE) and the Phasor Measurement Units (PMU’s) network constitute the communication infrastructures meant to provide the “smart grid” dispatcher with wide-area bus phase angles and other data, from which the operational status of the grid can be assessed— if the measurements are not compromised somewhere along their way to the ...

The \arrowhead torus" is a broadcast graph that we deene on the 6-valent grid as a Cayley graph. We borrow the term from Mandelbrot who qualiies in that way one of the Sierpinski's famous fractal constructions. The 6-valent grid H = (V; E) is generated by three families of straight lines. We adopt the isotropic orientation S ! N, NE ! SW, NW ! SE and deene the system of generators S = fs 1 ; s ...

We research on the possible orientations patterns of a grid graph G, and propose a method for counting certain combinatorial structures over the class of orientations of G. For example, our method can be applied for counting sink-free orientations of G, as well as it can be applied for solving the #2SAT problem for grid Boolean formulas. Our proposal extends the classical transfer matrix method...

A convex grid drawing of a plane graph G is a drawing of G on the plane so that all vertices of G are put on grid points, all edges are drawn as straight-line segments between their endpoints without any edge-intersection, and every face boundary is a convex polygon. In this paper we give a linear-time algorithm for finding a convex grid drawing of any 4-connected plane graph G with four or mor...