A Combinatorial Decomposition Theory

Abstract

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 { E 1 , E 2 } of E ( G ), W(G; E 1 ) denotes the set V(G(E 1 )) ⋂ V ( G ( E 2 )). We say that G is 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 G is a partition { E 1 , E 2 } of E(G) such that | E 1 | ≧ 2 ≧ |E 2 | and | W(G; E 1 ) | = 2. Where { E 1 , E 2 } is a split of G, W(G; E 2 ) = { u, v }, and e is an element not in E(G), we form graphs G i i = 1 and 2, by adding e to G(E i ) as an edge joining u to v.