On uniqueness of a general factorization of graph properties

Abstract

A graph property is any class of simple graphs, which is closed under isomorphisms. Let H be a given graph on vertices v1, …, vn. For graph properties 𝒫1, …, 𝒫n, we denote by H[𝒫1, …, 𝒫n] the class of those (𝒫1, …, 𝒫n) -partitionable graphs G, with a corresponding vertex partition (V1, …, Vn), for which an edge {xi, xj} with xi∈Vi and xj∈Vj implies the existence of the edge {vi, vj} in the graph H. The problem of the unique description of a graph property 𝒫 in the form H[𝒫1, …, 𝒫n] is investigated for 𝒫, 𝒫1, …, 𝒫n being from the class La of all graph properties closed under taking disjoint unions and subgraphs. The unique factorization theorems obtained in the paper generalize known results of this type bringing together ∘ -reducibility over La and ∨ -reducibility in the lattice (La, ⊆). There is also offered a new insight into the modular decomposition tree for a graph. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 48–64, 2009