Showing papers on "K-tree published in 1972"

Algorithms for Minimum Coloring, Maximum Clique, Minimum Covering by Cliques, and Maximum Independent Set of a Chordal Graph

Abstract: A finite undirected graph is called chordal if every simple circuit has a chord. Given a chordal graph, we present, ways for constructing efficient algorithms for finding a minimum coloring, a minimum covering by cliques, a maximum clique, and a maximum independent set. The proofs are based on a theorem of D. Rose [3] that a finite graph is chordal if and only if it has some special orientation called an R-orientation. In the last part of this paper we prove that an infinite graph is chordal if and only if it has an R-orientation.

Equivalent Formulations of the Hirsch Conjecture for Abstract Polytopes

Abstract: : polytopes are mathematical creations which are defined by three axioms. It has been shown that simple polytopes are a proper subclass of abstract polytopes. Hence theorems proving facts about abstract polytopes in general, prove facts about simple polytopes in particular. Klee and Walkup showed the following four statements were mathematically equivalent for simple polytopes: Any two vertices of a simple polytope can be joined by a (W sub v) (nonreturning) path. Delta(n,d) < or = n - d (Hirsch conjecture). Delta(2d,d) < or = d . For a Dantzig figure, (P,x,y) , delta sub p (x,y) = d . The purpose of the paper is to show that the four statements above are equivalent for the larger class of abstract polytopes as well.