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A Combinatorial Decomposition Theory

Published online by Cambridge University Press:  20 November 2018

William H. Cunningham  and
Jack Edmonds
William H. Cunningham
Affiliation:
Carleton University, Ottaway Ontario
Jack Edmonds
Affiliation:
University of Waterloo, Waterloo, Ontario
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Given a finite undirected graph G and A ⊆ E(G), G(A) denotes the subgraph of G having edge-set A and having no isolated vertices. For a partition {E1, E2} of E(G), W(G; E1) denotes the set V(G(E1)) ⋂ V(G(E2)). We say that Gis non-separable if it is connected and for every proper, non-empty subset A of E(G), we have |W(G; A)| ≧ 2. A split of a non-separable graph Gis a partition {E1, E2} of E(G) such that

|E1| ≧ 2 ≧ |E2| and |W(G; E1)| = 2.

Where {E1, E2} is a split of G, W(G; E2) = {u, v}, and e is an element not in E(G), we form graphs Gii= 1 and 2, by adding e to G(Ei) as an edge joining u to v.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 3 , 01 June 1980 , pp. 734 - 765
Copyright
Copyright © Canadian Mathematical Society 1980

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