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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
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals
Published: Nov 17, 2008
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