On subgraphs of Cartesian product graphs and S-primeness

AbstractIn 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

Maximal proper subgraphs of median graphs

AbstractFor a median graph G and a vertex v of G that is not a cut-vertex we show that G-v is a median graph precisely when v is not the center of a bipartite wheel, which is in turn equivalent with the existence of a certain edge elimination scheme for edges incident with v. This implies a characterization of vertex-critical (respectively, vertex-complete) median graphs, which are median graphs whose all vertex-deleted subgraphs are not median (respectively, are median). Moreover, two analogous characterizations for edge-deleted median graphs are given