Unique factorization of compositive hereditary graph properties

A graph property (i.e., a set of graphs) is induced-hereditary or additive if it is closed under taking induced-subgraphs or disjoint unions. If $\cP$ and $\cQ$ are properties, the product $\cP \circ \cQ$ consists of all graphs $G$ for which there is a partition of the vertex set of $G$ into (possibly empty) subsets $A$ and $B$ with $G[A] \in \cP$ and $G[B] \in \cQ$. A property is reducible if it is the product of two other properties, and irreducible otherwise. We completely describe the few reducible induced-hereditary properties that have a unique factorisation into irreducibles. Analogs of compositive and additive induced-hereditary properties are introduced and characterised in the style of Scheinerman [{\em Discrete Math}. {\bf 55} (1985) 185--193]. One of these provides an alternative proof that an additive hereditary property factors into irreducible additive hereditary properties.

Discussiones Mathematicae Graph Theory, 1995

Let IL be the set of all hereditary and additive properties of graphs. For P 1 , P 2 ∈ IL, the reducible property R = P 1 • P 2 is defined as follows: G ∈ R if and only if there is a partition V (G) = V 1 ∪ V 2 of the vertex set of G such that V 1 G ∈ P 1 and V 2 G ∈ P 2. The aim of this paper is to investigate the structure of the reducible properties of graphs with emphasis on the uniqueness of the decomposition of a reducible property into irreducible ones.

2000

A property P of infinite graphs is said to be of finite character if a graph G has property P if and only if every finite vertex-induced subgraph of G has property P. Using a generalization of the well-known Erd˝os-de Bruijn Theo- rem for arbitrary properties of finite character, we present a proof of the Unique Factorization Theorem for additive

European Journal of Combinatorics, 2011

v n }, E) replacing every vertex v i of H by the graph G i and joining the vertices of G i with all the vertices of those of G j whenever {v i , v j } ∈ E(H). For unlabelled graphs G 1 ,. .. , G n , H, let ϕ H (G 1 ,. .. , G n) stand for the class of all graphs H[G 1 ,. .. , G n ] taken over all possible orderings of V (H). A prime inductive class of graphs, I(B, C), is said to be a set of all graphs, which can be produced by recursive applying of ϕ H (G 1 ,. .. , G |V (H)|) where H is a graph from a fixed set C of prime graphs and G 1 ,. .. , G |V (H)| are either graphs from the set B of prime graphs or graphs obtained in the previous steps. Similar inductive definitions for cographs, k-trees, series-parallel graphs, Halin graphs, bipartite cubic graphs or forbidden structures of some graph classes were considered in the literature (Batagelj (1994) [1] Drgas-Burchardt et al. (2010) [6] and Hajós (1961) [10]). This paper initiates a study of prime inductive classes of graphs giving a result, which characterizes, in their language, the substitution closed induced hereditary graph classes. Moreover, for an arbitrary induced hereditary graph class P it presents a method for the construction of maximal induced hereditary graph classes contained in P and substitution closed. The main contribution of this paper is to give a minimal forbidden graph characterization of induced hereditary prime inductive classes of graphs. As a consequence, the minimal forbidden graph characterization for some special induced hereditary prime inductive graph classes is given There is also offered an algebraic view on the class of all prime inductive classes of graphs of the type I({K 1 }, C).

Journal of Graph Theory, 2008

A (g, f)-factor of a graph is a subset F of E such that for all v ∈ V, g(v) ≤ deg F (v) ≤ f(v). Lovasz gave a necessary and sufficient condition for the existence of a (g, f)-factor. We extend, to the case of edge-weighted graphs, a result of Kano and Saito who showed that if g(v) < λdeg E (v) < f(v) for any λ ∈ [0, 1], then a (g, f)-factor always exist. In addition, we use results of Anstee to provide new necessary and sufficient conditions for the existence of a (g, f)-factor.

2003

We show that additive induced-hereditary properties of coloured hypergraphs can be uniquely factorised into irreducible factors. Our constructions and proofs are so general that they can be used for arbitrary concrete categories of combinatorial objects; we provide some examples of such combinatorial objects.

Discussiones Mathematicae Graph Theory, 1997

In this paper we survey results and open problems on the structure of additive and hereditary properties of graphs. The important role of vertex partition problems, in particular the existence of uniquely partitionable graphs and reducible properties of graphs in this structure, is emphasized. Many related topics, including questions on the complexity of related problems, are investigated.

The Electronic Journal of Combinatorics

Can the vertices of an arbitrary graph $G$ be partitioned into $A \cup B$, so that $G[A]$ is a line-graph and $G[B]$ is a forest? Can $G$ be partitioned into a planar graph and a perfect graph? The NP-completeness of these problems are special cases of our result: if ${\cal P}$ and ${\cal Q}$ are additive induced-hereditary graph properties, then $({\cal P}, {\cal Q})$-colouring is NP-hard, with the sole exception of graph $2$-colouring (the case where both ${\cal P}$ and ${\cal Q}$ are the set ${\cal O}$ of finite edgeless graphs). Moreover, $({\cal P}, {\cal Q})$-colouring is NP-complete iff ${\cal P}$- and ${\cal Q}$-recognition are both in NP. This completes the proof of a conjecture of Kratochvíl and Schiermeyer, various authors having already settled many sub-cases.

Indonesian Journal of Combinatorics

Let H and G be two simple graphs. The concept of an H-magic decomposition of G arises from the combination between graph decomposition and graph labeling. A decomposition of a graph G into isomorphic copies of a graph H is H-magic if there is a bijection f : V (G) ∪ E(G) -→ {1, 2, ..., |V (G)∪E(G)|} such that the sum of labels of edges and vertices of each copy of H in the decomposition is constant. A lexicographic product of two graphs G 1 and G 2 , denoted by G 1 [G 2 ], is a graph which arises from G 1 by replacing each vertex of G 1 by a copy of the G 2 and each edge of G 1 by all edges of the complete bipartite graph K n,n where n is the order of G 2 . In this paper we provide a sufficient condition for C n [K m ] in order to have a P t [K m ]-magic decompositions, where n > 3, m > 1, and t = 3, 4, n -2.

2006

A homogeneous factorisation(M,G,Γ,P) is a partition P of the arc set of a digraph Γ such that there exist vertex-transitive groups such that M fixes each part of P setwise while G acts transitively on P. Homogeneous factorisations of complete graphs have previously been studied by the second and fourth authors, and are a generalisation of vertex-transitive self-complementary digraphs. In this paper we initiate the study of homogeneous factorisations of arbitrary graphs and digraphs.