Bidirected minimum Manhattan network problem

In the bidirected minimum Manhattan network problem, given a set T of n terminals in the plane, we need to construct a network N(T) of minimum total length with the property that the edges of N(T) are axis-parallel and oriented in a such a way that every ordered pair of terminals is connected in N(T) by a directed Manhattan path. In this paper, we present a polynomial factor 2 approximation algorithm for the bidirected minimum Manhattan network problem.Comment: 14 pages, 16 figure

The Manhattan product of digraphs

We give a formal definition of a new product of bipartite digraphs, the Manhattan product, and we study some of its main properties. It is shown that when all the factors of the above product are (directed) cycles, then the obtained digraph is the Manhattan street network. To this respect, it is proved that many properties of such networks, such as high symmetries and the presence of Hamiltonian cycles, are shared by the Manhattan product of some digraphs

The Minimum Shared Edges Problem on Grid-like Graphs

We study the NP-hard Minimum Shared Edges (MSE) problem on graphs: decide whether it is possible to route $p$ paths from a start vertex to a target vertex in a given graph while using at most $k$ edges more than once. We show that MSE can be decided on bounded (i.e. finite) grids in linear time when both dimensions are either small or large compared to the number $p$ of paths. On the contrary, we show that MSE remains NP-hard on subgraphs of bounded grids. Finally, we study MSE from a parametrised complexity point of view. It is known that MSE is fixed-parameter tractable with respect to the number $p$ of paths. We show that, under standard complexity-theoretical assumptions, the problem parametrised by the combined parameter $k$, $p$, maximum degree, diameter, and treewidth does not admit a polynomial-size problem kernel, even when restricted to planar graphs

The spectra of Manhattan street networks

AbstractThe multidimensional Manhattan street networks constitute a family of digraphs with many interesting properties, such as vertex symmetry (in fact they are Cayley digraphs), easy routing, Hamiltonicity, and modular structure. From the known structural properties of these digraphs, we determine their spectra, which always contain the spectra of hypercubes. In particular, in the standard (two-dimensional) case it is shown that their line digraph structure imposes the presence of the zero eigenvalue with a large multiplicity

The multidimensional Manhattan networks

The $n$-dimensional Manhattan network $M_n$---a special case of $n$-regular digraph---is formally defined and some of its structural properties are studied. In particular, it is shown that $M_n$ is a Cayley digraph, which can be seen as a subgroup of the $n$-dim version of the wallpaper group $pgg$. These results induce a useful new presentation of $M_n$, which can be applied to design a (shortest-path) local routing algorithm and to study some other metric properties. Also it is shown that the $n$-dim Manhattan networks are Hamiltonian and, in the standard case (that is, dimension two), they can be decomposed in two arc-disjoint Hamiltonian cycles. Finally, some results on the connectivity and distance-related parameters of $M_n$, such as the distribution of the node  distances and the diameter are presented

The spectra of Manhattan street networks

The multidimensional Manhattan street networks constitute a family of digraphs with many interesting properties, such as vertex symmetry (in fact they are Cayley digraphs), easy routing, Hamiltonicity, and modular structure. From the known structural properties of these digraphs, we determine their spectra, which always contain the spectra of hypercubes. In particular, in the standard (two-dimensional) case it is shown that their line digraph structure imposes the presence of the zero eigenvalue with a large multiplicity