Unique factorisation of additive induced-hereditary properties

Abstract

An additive hereditary graph property is a set of graphs, closed under isomorphism and under taking subgraphs and disjoint unions. Let ${\mathcal{P}}_1,>...,{\mathcal{P}}_n$ be additive hereditary graph properties. A graph $G$ has property $({\mathcal{P}}_1\circ ...\circ {\mathcal{P}}_n)$ if there is a partition $(V_1,...,V_n)$ of $V(G)$ into $n$ sets such that, for all $i$, the induced subgraph $G[V_i]$ is in ${\mathcal{P}}_i$. A property $\mathcal{P}$ is reducible if there are properties $\mathcal{Q}$, $\mathcal{R}$ such that $\mathcal{P}=\mathcal{Q}\circ \mathcal{R}$; otherwise it is irreducible. Mihók, Semanišin and Vasky [J. Graph Theory {\bf 33} (2000), 44--53] gave a factorisation for any additive hereditary property $\mathcal{P}$ into a given number $dc(\mathcal{P})$ of irreducible additive hereditary factors. Mihók [Discuss. Math. Graph Theory {\bf 20} (2000), 143--153] gave a similar factorisation for properties that are additive and induced-hereditary (closed under taking induced-subgraphs and disjoint unions). Their results left open the possiblity of different factorisations, maybe even with a different number of factors; we prove here that the given factorisations are, in fact, unique.

26 pages, 4 figures, to appear in Discussiones Mathematicae Graph Theory