3,447 research outputs found
Geodetic topological cycles in locally finite graphs
We prove that the topological cycle space C(G) of a locally finite graph G is
generated by its geodetic topological circles. We further show that, although
the finite cycles of G generate C(G), its finite geodetic cycles need not
generate C(G).Comment: 1
Convexity in partial cubes: the hull number
We prove that the combinatorial optimization problem of determining the hull
number of a partial cube is NP-complete. This makes partial cubes the minimal
graph class for which NP-completeness of this problem is known and improves
some earlier results in the literature.
On the other hand we provide a polynomial-time algorithm to determine the
hull number of planar partial cube quadrangulations.
Instances of the hull number problem for partial cubes described include
poset dimension and hitting sets for interiors of curves in the plane.
To obtain the above results, we investigate convexity in partial cubes and
characterize these graphs in terms of their lattice of convex subgraphs,
improving a theorem of Handa. Furthermore we provide a topological
representation theorem for planar partial cubes, generalizing a result of
Fukuda and Handa about rank three oriented matroids.Comment: 19 pages, 4 figure
Algorithms to Find Linear Geodetic Numbers and Linear Edge Geodetic Numbers in Graphs
Given two vertices u and v of a connected graph G=(V, E), the closed interval I[u, v] is that set of all vertices lying in some u-v geodesic in G. A subset of V(G) S={v1,v2,v3,….,vk} is a linear geodetic set or sequential geodetic set if each vertex x of G lies on a vi – vi+1 geodesic where 1 ? i < k . A linear geodetic set of minimum cardinality in G is called as linear geodetic number lgn(G) or sequential geodetic number sgn(G). Similarly, an ordered set S={v1,v2,v3,….,vk} is a linear edge geodetic set if for each edge e = xy in G, there exists an index i, 1 ? i < k such that e lies on a vi – vi+1 geodesic in G. The cardinality of the minimum linear edge geodetic set is the linear edge geodetic number of G denoted by legn(G). The purpose of this paper is to introduce algorithms using dynamic programming concept to find minimum linear geodetic set and thereby linear geodetic number and linear edge geodetic set and number in connected graphs
On metric Ramsey-type phenomena
The main question studied in this article may be viewed as a nonlinear
analogue of Dvoretzky's theorem in Banach space theory or as part of Ramsey
theory in combinatorics. Given a finite metric space on n points, we seek its
subspace of largest cardinality which can be embedded with a given distortion
in Hilbert space. We provide nearly tight upper and lower bounds on the
cardinality of this subspace in terms of n and the desired distortion. Our main
theorem states that for any epsilon>0, every n point metric space contains a
subset of size at least n^{1-\epsilon} which is embeddable in Hilbert space
with O(\frac{\log(1/\epsilon)}{\epsilon}) distortion. The bound on the
distortion is tight up to the log(1/\epsilon) factor. We further include a
comprehensive study of various other aspects of this problem.Comment: 67 pages, published versio
- …