Bisubmodular polyhedra, simplicial divisions, and discrete convexity

Bisubmodular polyhedra, simplicial divisions, and discrete convexity

We consider a class of integer-valued discrete convex functions, called BS-convex functions, defined on integer lattices whose affinity domains are sets of integral points of integral bisubmodular polyhedra. We examine discrete structures of BSconvex functions and give a characterization of BS-convex functions in terms of their convex conjugate functions by means of (discordant) Freudenthal sim...

Delta-Matroids, Jump Systems, and Bisubmodular Polyhedra

We relate an axiomatic generalization of matroids, called a jump system, to poly-hedra arising from bisubmodular functions. Unlike the case for usual submodularity, the points of interest are not all the integral points in the relevant polyhedron, but form a subset of them. However, we do show that the convex hull of the set of points of a jump system is a bisubmodular polyhedron, and that the ...

Neighbor Systems, Jump Systems, and Bisubmodular Polyhedra

The concept of neighbor system, introduced by Hartvigsen (2009), is a set of integral vectors satisfying a certain combinatorial property. In this paper, we reveal the relationship of neighbor systems with jump systems and with bisubmodular polyhedra. We firstly prove that for every neighbor system, there exists a jump system which has the same neighborhood structure as the original neighbor sy...

Vertex-Unfoldings of Simplicial Polyhedra

We present two algorithms for unfolding the surface of any polyhedron, all of whose faces are triangles, to a nonoverlapping, connected planar layout. The surface is cut only along polyhedron edges. The layout is connected, but it may have a disconnected interior: the triangles are connected at vertices, but not necessarily joined along edges.

Convexity recognition of the union of polyhedra