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Factoring cartesian‐product graphs
Author(s) -
Imrich Wilfried,
Žerovnik Janez
Publication year - 1994
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.3190180604
Subject(s) - mathematics , cartesian product , combinatorics , equivalence relation , discrete mathematics , prime factor , graph product , graph , chordal graph , prime (order theory) , 1 planar graph
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.