2,905 research outputs found
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
Convex Cycle Bases
Convex cycles play a role e.g. in the context of product graphs. We introduce convex cycle bases and describe a polynomial-time algorithm that recognizes whether a given graph has a convex cycle basis and provides an explicit construction in the positive case. Relations between convex cycles bases and other types of cycles bases are discussed. In particular we show that if G has a unique minimal cycle bases, this basis is convex. Furthermore, we characterize a class of graphs with convex cycles bases that includes partial cubes and hence median graphs. (authors' abstract)Series: Research Report Series / Department of Statistics and Mathematic
Hypercellular graphs: partial cubes without as partial cube minor
We investigate the structure of isometric subgraphs of hypercubes (i.e.,
partial cubes) which do not contain finite convex subgraphs contractible to the
3-cube minus one vertex (here contraction means contracting the edges
corresponding to the same coordinate of the hypercube). Extending similar
results for median and cellular graphs, we show that the convex hull of an
isometric cycle of such a graph is gated and isomorphic to the Cartesian
product of edges and even cycles. Furthermore, we show that our graphs are
exactly the class of partial cubes in which any finite convex subgraph can be
obtained from the Cartesian products of edges and even cycles via successive
gated amalgams. This decomposition result enables us to establish a variety of
results. In particular, it yields that our class of graphs generalizes median
and cellular graphs, which motivates naming our graphs hypercellular.
Furthermore, we show that hypercellular graphs are tope graphs of zonotopal
complexes of oriented matroids. Finally, we characterize hypercellular graphs
as being median-cell -- a property naturally generalizing the notion of median
graphs.Comment: 35 pages, 6 figures, added example answering Question 1 from earlier
draft (Figure 6.
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