For a compact, orientable, irreducible, ∂-irreducible, and an-annular 3-manifold, it is shown there are only finitely many boundary slopes for incompressible and ∂-incompressible surfaces of a bounded Euler characteristic. We use normal surface theory and the inverse relationship of crushing a triangulation along a normal surface [8] and that of inflating an ideal triangulation [12] to introduce and study boundary-efficient triangulations and end-efficient ideal triangulations. It is shown for a compact 3-manifold with boundary, satisfying these topological conditions, any triangulation can be modified to a boundary-efficient triangulation; furthermore, it can be decided if a triangulation of such a manifold is boundary-efficient.

对于一个紧凑的，可定向的，不可约的，∂不可约的和环形的三歧管，表明有限的欧拉特征的不可压缩和∂不可压缩表面只有有限的边界斜率。我们使用法向曲面理论和沿法向曲面压碎三角剖分与使理想三角剖分膨胀的逆关系[12]引入和研究边界有效三角剖分和最终有效理想三角剖分。它显示为具有边界的紧凑型3流形，满足这些拓扑条件，任何三角剖分都可以修改为边界有效的三角剖分。此外，可以确定这种歧管的三角剖分是否是边界有效的。