Factoring cartesian‐product graphs

Abstract

In a fundamental paper, G. Sabidussi [“Graph Multiplication,” Mathematische Zeitschrift, Vol. 72 (1960), pp. 446–457] used a tower of equivalence relations on the edge set E(G) of a connected graph G to decompose G into a Cartesian product of prime graphs. Later, a method by R.L. Graham and P.M. Winkler [“On Isometric Embeddings of Graphs,” Transactions of the American Mathematics Society, Vol. 288 (1985), pp. 527–533] of embedding a connected graph isometrically into Cartesian products opened another approach to this problem. In both approaches an equivalence relation σ that determines the prime factorization is constructed. The methods differ by the starting relations used. We show that σ can be obtained as the convex hull of the starting relation used by Sabidussi. Our result also holds for the relation determining the prime decomposition of infinite connected graphs with respect to the weak Cartesian product. Moreover, we show that this relation is the transitive closure of the union of the starting relations of Sabidussi and Winkler [“Factoring a Graph in Polynomial Time,” European Journal of Combinatorics, Vol. 8 (1987), pp. 209–212], thereby generalizing the result of T. Feder [“Product Graph Representations,” Journal of Graph Theory, Vol 16 (1993), pp. 467–488] from finite to infinite graphs.

The Cartesian product of hypergraphs

“…A PFD of H is a representation as a Cartesian product H = i∈I H i , or as a weak Cartesian product H = a i∈I H i , resp., such that all factors H i , i ∈ I, are prime and H i K 1 . In order to show that every hypergraph has a unique representation as a weak Cartesian product, we follow the strategy of Imrich andŽerovnik [14] and characterize the so-called product relations defined on the arc set E of H. The advantage of this approach is that it does not require finiteness and hence also pertains to the weak Cartesian product of infinitely many factors.…”

“…Here we extend and generalize these results further and show that every connected directed hypergraph has a unique prime factor decomposition (PFD) with respect to the (weak) Cartesian product. Instead of following the proof strategies of the classical papers, we adopt the approach of Imrich andŽerovnik [14] that constructs a product relation starting from simpler relations on the edge set E. In the case of graphs, the square-property [12] plays a central role as technical device. The grid-property, which is introduced here, serves as generalization of this construction.…”

Graph homomorphisms: structure and symmetry

“…A similar treatment applies to isomorphisms into cartesian products, which can be viewed as a particular kind of isometric embeddings (see [83]). However, we shall not go into this subject and turn our attention to the retracts of a cartesian product of graphs.…”

On Normal Cayley Graphs and Hom-idempotent Graphs

“…Albertson and Collins [1] first proved the contrasting result that if P denotes the Petersen graph, then for any integers m > n, there does not exist a homomorphism from P m to P n . Their method used the 'No-Homomorphism Lemma', thus involved a computation of the independence number of P n for all n. Later, Zhou [13] computed the independence number of cartesian powers of any circulant G by using the existing homomorphisms from G n+1 to G n for all n. This line of research was pursued by Hell, Yu and Zhou [11]; they called a graph G 'hom-regular' if there exists a homomorphism from G 2 to G. On this subject, Hahn, Hell and Poljak [8] posed a question that inspired our work: let G be a Cayley graph such that for some integer n, G n+1 admits a homomorphism to G n . Does this already imply that there exists a homomorphism from G 2 to G?…”

“…We begin by recalling some results of Imrich andZerovnik [13] on this subject. Let G be a connected graph.…”