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**Factoring Cartesian-product graphs.**
*(English)*
Zbl 0811.05054

The authors consider the structure of the equivalence relation \(\sigma\) on the edge set of a graph \(G\) belonging to the decomposition of \(G\) into a Cartesian product of prime graphs; this prime factorization theory was founded by G. Sabidussi [Math. Z. 72, 446-457 (1960; Zbl 0093.376)]. For a family of graphs \((G_ \iota)_{\iota \in I}\), the Cartesian product \(\Gamma = \square_{\iota \in I} G_ \iota\) is the graph \(\Gamma = (V(\Gamma), E(\Gamma))\) with \(V(\Gamma) := \{(x_ \iota)_{\iota \in I} : x_ \iota \in V(G_ \iota)\), \(\iota \in I\}\) and \(E(\Gamma) := \bigcup_{\kappa \in I} E_ \kappa\), where \(E_ \kappa := \{\{(x_ \iota), (y_ \iota)\} : (x_ \iota), (y_ \iota) \in V(\Gamma) \wedge \{x_ \kappa, y_ \kappa\} \in E(G_ \kappa) \wedge \forall \iota (\iota \in I - \{\kappa\} \to x_ \iota = y_ \iota)\}\), \(\kappa \in I\); for any \(a = (a_ \iota) \in V(\Gamma)\), the connected component \(G\) of \(\Gamma\) containing \(a\) is called the weak Cartesian product \(G = \square^ a_{\iota \in I} G_ \iota\). The decomposition \(E(G) = \bigcup_{\kappa \in I}(E_ \kappa \cap E(G))\) determines an equivalence relation \(\sigma(\square^ a G_ \iota)\) on \(E(G)\) belonging to this weak Cartesian product. For any connected graph \(G\), it is defined (a) \(e, f \in E(G)\) are in the relation \(\delta\) iff either \(e\) and \(f\) are adjacent and there is no chordless square containing \(e\) and \(f\) or \(e = f\) or \(e\) and \(f\) are opposite edges of a chordless square; (b) \(e = \{x, y\} \in E(G)\) and \(f = \{x', y'\} \in E(G)\) are in (Djoković’s) relation \(\Theta\) iff the distances satisfy \(d(x,x') + d(y,y') \neq d(x,y') + d(x', y)\); (c) a subgraph \(H\) of \(G\) is convex (in \(G\)) iff every shortest \(G\)-path between any two vertices of \(H\) is in \(H\); (d) an equivalence relation \(\gamma\) on \(E(G)\) with the equivalence classes \(E_ \iota\), \(\iota \in I\), is convex iff for any \(K \subseteq I\) every connected component of the subgraph induced on \(\bigcup_{\iota \in K} E_ \iota\) is convex; (e) \(e = \{x, z\} \in E(G)\) and \(f = \{z, y\} \in E(G)\) are in the relation \(\tau\) iff \(z\) is the unique common neighbour of \(x\) and \(y\). Obviously, for any weak Cartesian product representation \(\square^ a _{\iota \in I} G_ \iota\) of \(G\) the relation \(\sigma(\square^ a G_ \iota)\) is convex and contains the transitive closure \(\delta^*\) of \(\delta\).

Now, the following main results are proved for any (finite or infinite) connected graph \(G\): (1) Every convex equivalence relation \(\gamma \supseteq \delta\) on \(E(G)\) induces a representation of \(G\) as a weak Cartesian product. (2) The intersection of an arbitrary set of convex equivalence relations on \(E(G)\) containing \(\delta\) is convex. Thus there is a finest convex equivalence relation on \(E(G)\) containing \(\delta\), the convex hull \(C(\delta)\) of \(\delta\). (3) From (1) and (2) it follows a new proof of the known statement that \(G\) has a unique representation as a weak Cartesian product of prime graphs (prime factorization). (4) The equivalence relation \(\sigma\) on \(E(G)\) belonging to this prime factorization is \(\sigma = C(\delta)\). (5) \(\sigma = (\delta \cup \Theta)^*\); moreover, every convex equivalence relation \(\gamma \supseteq \tau\) on \(E(G)\) contains \(\delta\), and \(\sigma = (\tau \cup \Theta)^* = C(\tau)\).

Now, the following main results are proved for any (finite or infinite) connected graph \(G\): (1) Every convex equivalence relation \(\gamma \supseteq \delta\) on \(E(G)\) induces a representation of \(G\) as a weak Cartesian product. (2) The intersection of an arbitrary set of convex equivalence relations on \(E(G)\) containing \(\delta\) is convex. Thus there is a finest convex equivalence relation on \(E(G)\) containing \(\delta\), the convex hull \(C(\delta)\) of \(\delta\). (3) From (1) and (2) it follows a new proof of the known statement that \(G\) has a unique representation as a weak Cartesian product of prime graphs (prime factorization). (4) The equivalence relation \(\sigma\) on \(E(G)\) belonging to this prime factorization is \(\sigma = C(\delta)\). (5) \(\sigma = (\delta \cup \Theta)^*\); moreover, every convex equivalence relation \(\gamma \supseteq \tau\) on \(E(G)\) contains \(\delta\), and \(\sigma = (\tau \cup \Theta)^* = C(\tau)\).

Reviewer: G.Schaar (Freiberg)

### MSC:

05C70 | Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) |

05C99 | Graph theory |

05C12 | Distance in graphs |

### Keywords:

equivalence relation; Cartesian product; prime graphs; prime factorization; distances; convex hull### Citations:

Zbl 0093.376
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\textit{W. Imrich} and \textit{J. Žerovnik}, J. Graph Theory 18, No. 6, 557--567 (1994; Zbl 0811.05054)

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### References:

[1] | Aurenhammer, Cartesian Graph Factorization Comput, Complexity 2 pp 331– (1992) |

[2] | Aurenhammer, Computing equivalence classes among the edges of a graph with applications, Discrete Math. 109 pp 3– (1992) · Zbl 0795.05130 |

[3] | Feder, Product graph representations, J. Graph Theory. 16 pp 467– (1993) · Zbl 0766.05092 |

[4] | Feigenbaum, A polynomial time algorithm for finding the prime factors of Cartesian-product graphs, Discrete Appl. Math. 12 pp 123– (1985) · Zbl 0579.68028 |

[5] | Graham, On isometric embeddings of graphs, Trans. Am. Math. Soc. 288 pp 527– (1985) · Zbl 0576.05017 |

[6] | Hochstrasser, A note on Winkler’s algorithm for factoring a connected graph, Discrete Math. 109 pp 127– (1992) · Zbl 0780.05045 |

[7] | Imrich, Über das schwache Kartesische Produkt von Graphen, J. Combinat. Theory Ser. B 11 pp 1– (1971) · Zbl 0218.05069 |

[8] | Imrich, Embedding graphs into cartesian products, Ann. NY Acad. Sci. 576 pp 266– (1989) · Zbl 0792.05044 |

[9] | Miller, Weak Cartesian product of graphs, Colloq. Math. 21 pp 55– (1970) · Zbl 0195.54301 |

[10] | Sabidussi, Graph multiplication, Math. Z. 72 pp 446– (1960) · Zbl 0093.37603 |

[11] | C. Tardif Prefibres in Cartesian product of metric spaces University of Montreal |

[12] | Vizing, The Cartesian product of graphs [in Russian]., Vyčisl. Syst. 9 pp 30– (1963) |

[13] | Comp. El. Syst. 2 1966 352 365 |

[14] | Winkler, Factoring a graph in polynomial time, Eur. J. Combinat. 8 pp 209– (1987) · Zbl 0625.05050 |

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