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On rigid circuit graphs

On rigid circuit graphs By G. A. DmAc (Hamburg) 1. Introduction The graphs considered in this paper may have loops (German: Schlinge) and multiple edges and they may be infinite, except where the contrary is stated. The axiom of choice is assumed throughout. Definitions 1. If/2 is a circuit and x and y are two distinct vertices of/2 which are not joined by any edge belonging to/2, then an edge which joins x and y is called a chord of/2. A graph in which every circuit with more than three vertices has at least one chord, is called a rigid circuit graph, or ri. cir. graph for short. All trees and all cliques (see below) are ri. cir. graphs, and so are the interval graphs investigated by G. HAJ6s [2]. Theorems on ri. cir. graphs have been given by A. HAJ~AL and J. SUR~YI [3], and by C. BE~o~ [1]. In the present paper a new characterisation of ri. cir. graphs will be established (Theorem 1) and the results of the above mentioned authors, and some new results, will be simply derived. Definitions 2. IfF is a connected graph and ~ is a set of vertices contain- ed in F, then ~ is http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

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References (4)

Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general; Algebra; Differential Geometry; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/BF02992776
Publisher site
See Article on Publisher Site

Abstract

By G. A. DmAc (Hamburg) 1. Introduction The graphs considered in this paper may have loops (German: Schlinge) and multiple edges and they may be infinite, except where the contrary is stated. The axiom of choice is assumed throughout. Definitions 1. If/2 is a circuit and x and y are two distinct vertices of/2 which are not joined by any edge belonging to/2, then an edge which joins x and y is called a chord of/2. A graph in which every circuit with more than three vertices has at least one chord, is called a rigid circuit graph, or ri. cir. graph for short. All trees and all cliques (see below) are ri. cir. graphs, and so are the interval graphs investigated by G. HAJ6s [2]. Theorems on ri. cir. graphs have been given by A. HAJ~AL and J. SUR~YI [3], and by C. BE~o~ [1]. In the present paper a new characterisation of ri. cir. graphs will be established (Theorem 1) and the results of the above mentioned authors, and some new results, will be simply derived. Definitions 2. IfF is a connected graph and ~ is a set of vertices contain- ed in F, then ~ is

Journal

Abhandlungen aus dem Mathematischen Seminar der Universität HamburgSpringer Journals

Published: Nov 17, 2008

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