On subgraphs of Cartesian product graphs and S-primeness

In this paper we consider S-prime graphs, that is the graphs that cannot be represented as nontrivial subgraphs of nontrivial Cartesian products of graphs. Lamprey and Barnes characterized S-prime graphs via so-called basic S-prime graphs that form a subclass of all S-prime graphs. However, the stru...

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Bibliographic Details
Published inDiscrete mathematics Vol. 282; no. 1; pp. 43 - 52
Main Author BRESAR, Bostjan
Format Journal Article
LanguageEnglish
Published Amsterdam Elsevier B.V 06.05.2004
Elsevier
Subjects
Algebra
Cartesian product
Combinatorics
Combinatorics. Ordered structures
Exact sciences and technology
Extension
Mathematics
Prime
Sciences and techniques of general use
Subgraph
Extension
Subgraph
05C75
05C99
Prime
Cartesian product
Characterization
Discrete mathematics
Graph product
Graph theory
Graph connectivity
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Summary:In this paper we consider S-prime graphs, that is the graphs that cannot be represented as nontrivial subgraphs of nontrivial Cartesian products of graphs. Lamprey and Barnes characterized S-prime graphs via so-called basic S-prime graphs that form a subclass of all S-prime graphs. However, the structure of basic S-prime graphs was not known very well. In this paper we prove several characterizations of basic S-prime graphs. In particular, the structural characterization of basic S-prime graphs of connectivity 2 enables us to present several infinite families of basic S-prime graphs. Furthermore, simple S-prime graphs are introduced that form a relatively small subclass of basic S-prime graphs, and it is shown that every basic S-prime graph can be obtained from a simple S-prime graph by a sequence of certain transformations called extensions.
ISSN:0012-365X
1872-681X
DOI:10.1016/j.disc.2003.11.005