电路直径和Klee-Walkup结构
Circuit diameter and Klee-Walkup constructions
摘要：
考虑一个多面体的图直径的变体，其中两个顶点之间的行走中的每一步最大限度地沿着回路方向而不是沿着入射边行进。这里的电路方向是非平凡的解决方案，以最小相关的子系统的介绍多面体。这些可以被理解为所有可能的边缘方向的集合，包括可能由小平面的平移产生的边缘。考虑图直径的赫希猜想的电路模拟是很有吸引力的，正如Borgwardt等人所建议的那样。[BFH 15]。他们问，赫希猜想的已知反例是否能为这个放松的回路直径概念提供反例。我们表明，最基本的反例无界赫希猜想，Klee-Walkup多面体，没有一个电路直径，满足赫希界，无论表示。我们还研究了有界Klee-Walkup多面体的回路直径。
英文摘要：
Consider a variant of the graph diameter of a polyhedron where each step in a walk between two vertices travels maximally in a circuit direction instead of along incident edges. Here circuit directions are non-trivial solutions to minimally-dependent subsystems of the presentation of the polyhedron. These can be understood as the set of all possible edge directions, including edges that may arise from translation of the facets. It is appealing to consider a circuit analogue of the Hirsch conjecture for graph diameter, as suggested by Borgwardt et al. [BFH15]. They ask whether the known counterexamples to the Hirsch conjecture give rise to counterexamples for this relaxed notion of circuit diameter. We show that the most basic counterexample to the unbounded Hirsch conjecture, the Klee-Walkup polyhedron, does have a circuit diameter that satisfies the Hirsch bound, regardless of representation. We also examine the circuit diameter of the bounded Klee-Walkup polytope.
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