Structure Functions and Minimal Path Sets

Number of Minimal Path Sets in a Consecutive-k-out-of-n: F System

In this paper the combinatorial problem of determining the number of minimal path sets of a consecutive-k-out-of-n: F system is considered. For the cases where k = 2, 3 the explicit formulae are given and for k ≥ 4 a recursive relation is obtained. Direct computation for determining the number of minimal path sets of a consecutive-k-out-of-n: F system for k ≥ 4 remains a difficult task. ...

Convex Sets and Minimal Sublinear Functions

We show that, given a closed convex set K with the origin in its interior, the support function of the set {y ∈ K∗ | ∃x ∈ K such that xy = 1} is the pointwise smallest sublinear function σ such that K = {x |σ(x) ≤ 1}.

The Structure of Minimal Sets

On the Structure and Complexity of Infinite Sets with Minimal Perfect Hash Functions

This paper studies the class of infinite sets that have minimal perfect hash functions­ one-to-one onto maps between the sets and E·-computable in polynomial time. We show that all standard NP-complete sets have polynomial-time computable minimal per­ fect hash functions, and give a structural condition sufficient to ensure that all infinite NP sets have polynomial-time computable minimal perfe...

Convex sets in graphs, II. Minimal path convexity

A set K of vertices in a connected graph is M-convex if and only if for every pair of vertices in K, all vertices of all chordless paths joining them also lie in K. Carathtodory, Helly and Radon type theorems are proved for M-convex sets. The Caratheodory number is 1 for complete graphs and 2 for other graphs, The Helly number equals the size of a maximum clique. The Radon number is one more th...