Theorem on directed graphs, applicable to logic

Abstract

A semicycle is said to turn at a point a if the arcs incident to a are both to it or both from it. We prove that if a nonempty set of points of a finite directed graph contains a turning point of each semicycle, then one of its members is a turning point of every semicycle to which it belongs; and we indicate the application of this result to mathematical logic through the modeling of arguments by graphs.We follow the terminology of Harary [l], and we assume in particular that a graph is finite and without loops or multiple arcs. We say also that a point a is a turning-point of a semiwalk. . . ,E, a, E', . . . if E and E' are both to a or both from a. Thus a walk is a semiwalk without turningpoints; and similarly with paths and cycles. Theorem 1. If a nonempty set S of points of a directed graph contains a turning point of each semicycle, then some member of S is a turning point of every semicycle to which it belongs.

Unifying Duality Theorems for Width Parameters in Graphs and Matroids (Extended Abstract)

“…We remark that while many cusped graphs are constructible [8], not all are. For example, a vertex m ∈ M in a constructible graph will never separate the other ends of two of its incident edges from the same partition class of E(m), but this can happen in an arbitrary cusped graph such as a tree.…”

“…If we delete the word 'weak' in this sentence, we obtain the consistent orientations of S k that avoid S * k = { S ∈ B k : S is a star}, which are precisely the dual objects to tree-decompositions of width < k − 1. 8 Our General Duality Theorem thus specializes to blocks as follows:…”