Retracts of Products of Chordal Graphs

In this article, we characterize the graphs G that are the retracts of Cartesian products of chordal graphs. We show that they are exactly the weakly modular graphs that do not contain K2,3, the 4-wheel minus one spoke W − 4 , and the k-wheels Wk (for k ≥ 4) as induced subgraphs. We also show that these graphs G are exactly the cage-amalgamation graphs as introduced by Brešar and Tepeh Horvat (Cage-amalgamation graphs, a common generalization of chordal and median graphs, Eur J Combin 30 (2009), 1071–1081); this solves the open question raised by these authors. Finally, we prove that replacing all products of cliques of G by products of Euclidean simplices, we obtain a polyhedral cell complex which, endowed Contract grant sponsor: French-Slovenian Egide PROTEUS project; Contract grant sponsor: Ministry of Science and Technology of Slovenia; Contract grant numbers: J1-2043 and P1-0297 (B. B. and M. K.); Contract grant sponsor: TEOMATRO; Contract grant number: ANR-10-BLAN 0207 (V. C.). Journal of Graph Theory C © 2012 Wiley Periodicals, Inc.