477 research outputs found
The Dilworth Number of Auto-Chordal-Bipartite Graphs
The mirror (or bipartite complement) mir(B) of a bipartite graph B=(X,Y,E)
has the same color classes X and Y as B, and two vertices x in X and y in Y are
adjacent in mir(B) if and only if xy is not in E. A bipartite graph is chordal
bipartite if none of its induced subgraphs is a chordless cycle with at least
six vertices. In this paper, we deal with chordal bipartite graphs whose mirror
is chordal bipartite as well; we call these graphs auto-chordal bipartite
graphs (ACB graphs for short). We describe the relationship to some known graph
classes such as interval and strongly chordal graphs and we present several
characterizations of ACB graphs. We show that ACB graphs have unbounded
Dilworth number, and we characterize ACB graphs with Dilworth number k
An identity theorem for the Fourier transform of polytopes on rationally parameterisable hypersurfaces
A set of points in is called a rationally
parameterisable hypersurface if , where is a vector function with domain and rational functions as
components. A generalized -dimensional polytope in is a union
of a finite number of convex -dimensional polytopes in . The
Fourier transform of such a generalized polytope in
is defined by . We prove that
implies if is an open subset of
satisfying some well-defined conditions. Moreover we show that this theorem can
be applied to quadric hypersurfaces that do not contain a line, but at least
two points, i.e., in particular to spheres.Comment: 20 page
Incremental Network Design with Minimum Spanning Trees
Given an edge-weighted graph and a set , the
incremental network design problem with minimum spanning trees asks for a
sequence of edges minimizing
where is the weight of a minimum spanning tree
for the subgraph and . We prove that this problem can be solved by a greedy
algorithm.Comment: 9 pages, minor revision based on reviewer comment
An asymptotic formula for the maximum size of an h-family in products of partially ordered sets
AbstractAn h-family of a partially ordered set P is a subset of P such that no h + 1 elements of the h-family lie on any single chain. Let S1, S2,… be a sequence of partially ordered sets which are not antichains and have cardinality less than a given finite value. Let Pn be the direct product of S1,…, Sn. An asymptotic formula of the maximum size of an h-family in Pn is given, where h=o(n) and n → ∞
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