13 research outputs found
Relationship of -Bend and Monotonic -Bend Edge Intersection Graphs of Paths on a Grid
If a graph can be represented by means of paths on a grid, such that each
vertex of corresponds to one path on the grid and two vertices of are
adjacent if and only if the corresponding paths share a grid edge, then this
graph is called EPG and the representation is called EPG representation. A
-bend EPG representation is an EPG representation in which each path has at
most bends. The class of all graphs that have a -bend EPG representation
is denoted by . is the class of all graphs that have a
monotonic (each path is ascending in both columns and rows) -bend EPG
representation.
It is known that holds for . We prove that
holds also for and for by investigating the -membership and -membership of complete
bipartite graphs. In particular we derive necessary conditions for this
membership that have to be fulfilled by , and , where and are
the number of vertices on the two partition classes of the bipartite graph. We
conjecture that holds also for .
Furthermore we show that holds for all
. This implies that restricting the shape of the paths can lead
to a significant increase of the number of bends needed in an EPG
representation. So far no bounds on the amount of that increase were known. We
prove that holds, providing the first result of this
kind
Monotonic Representations of Outerplanar Graphs as Edge Intersection Graphs of Paths on a Grid
A graph is called an edge intersection graph of paths on a grid if there
is a grid and there is a set of paths on this grid, such that the vertices of
correspond to the paths and two vertices of are adjacent if and only if
the corresponding paths share a grid edge. Such a representation is called an
EPG representation of . is the class of graphs for which there
exists an EPG representation where every path has at most bends. The bend
number of a graph is the smallest natural number for which
belongs to . is the subclass of containing all graphs
for which there exists an EPG representation where every path has at most
bends and is monotonic, i.e. it is ascending in both columns and rows. The
monotonic bend number of a graph is the smallest natural number
for which belongs to . Edge intersection graphs of paths on a
grid were introduced by Golumbic, Lipshteyn and Stern in 2009 and a lot of
research has been done on them since then.
In this paper we deal with the monotonic bend number of outerplanar graphs.
We show that holds for every outerplanar graph .
Moreover, we characterize in terms of forbidden subgraphs the maximal
outerplanar graphs and the cacti with (monotonic) bend number equal to ,
and . As a consequence we show that for any maximal outerplanar graph and
any cactus a (monotonic) EPG representation with the smallest possible number
of bends can be constructed in a time which is polynomial in the number of
vertices of the graph
New special cases of the quadratic assignment problem with diagonally structured coefficient matrices
We consider new polynomially solvable cases of the well-known Quadratic Assignment Problem involving coefficient matrices with a special diagonal structure. By combining the new special cases with polynomially solvable special cases known in the literature we obtain a new and larger class of polynomially solvable special cases of the QAP where one of the two coefficient matrices involved is a Robinson matrix with an additional structural property: this matrix can be represented as a conic combination of cut matrices in a certain normal form. The other matrix is a conic combination of a monotone anti-Monge matrix and a down-benevolent Toeplitz matrix. We consider the recognition problem for the special class of Robinson matrices mentioned above and show that it can be solved in polynomial time
Optimizing the Incidences between Points and Arcs on a Circle
Projet PRAXITELEGiven a set P of 2n+1 points regularly spaced on a circle, a number pi for pairwise distinct points and a number alpha for pairwise distinct and fixed length arcs incident to points in P, the sum of incidences between alpha arcs and pi points, is optimized by contiguously assigning both arcs and points. An extension to negative incidences by considering weights on points is provided. Optimizing a special case of a bilinear form (Hardy, Littlewood and PΓ³lya' theorem) as well as Circulant anti-Monge QAP directly follow
Travelling salesman paths on Demidenko matrices
In the path version of the Travelling Salesman Problem (Path-TSP), a salesman is looking for the shortest Hamiltonian path through a set of n cities. The salesman has to start his journey at a given city s, visit every city exactly once, and finally end his trip at another given city t.
In this paper we show that a special case of the Path-TSP with a Demidenko distance matrix is solvable in polynomial time. Demidenko distance matrices fulfill a particular condition abstracted from the convex Euclidian special case by Demidenko (1979) as an extension of an earlier work of Kalmanson (1975). We identify a number of crucial combinatorial properties of the optimal solution and design a dynamic programming approach with time complexity O(n6)
A Note on the Maximum of a certain Bilinear Form
In this note a generalization of a result by Hardy, Littlewood and P'olya (1926) is derived on computing the maximum of a certain bilinear form. The proof is elementary