A Combinatorial Decomposition Theory

Abstract

Given a finite undirected graphGandA⊆E(G),G(A)denotes the subgraph ofGhaving edge-setAand having no isolated vertices. For a partition {E1, E2}ofE(G),W(G; E1)denotes the setV(G(E1))⋂V(G(E2)). We say thatGisnon-separableif it is connected and for every proper, non-empty subsetAofE(G), we have |W(G;A)| ≧ 2. Asplitof a non-separable graphGis a partition {E1, E2} ofE(G)such that|E1| ≧ 2 ≧ |E2| and |W(G; E1)| = 2.Where {E1, E2} is a split ofG, W(G; E2)= {u, v}, andeis an element not inE(G),we form graphsGii= 1 and 2, by addingetoG(Ei)as an edge joiningutov.