Homogeneously Embedding Stratified Graphs in Stratified Graphs

A 2-stratified graph G is a graph whose vertex set has been partitioned into two subsets, called the strata or color classes of G. Two 2-stratified graphs G and H are isomorphic if there exists a color-preserving isomorphism φ from G to H. A 2-stratified graph G is said to be homogeneously embedded in a 2-stratified graph H if for every vertex x of G and every vertex y of H, where x and y are colored the same, there exists an induced 2-stratified subgraph H ′ of H containing y and a color-preserving isomorphism φ from G to H ′ such that φ(x) = y. A 2-stratified graph F of minimum order in which G can be homogeneously embedded is called a frame of G and the order of F is called the framing number fr(G) of G. It is shown that every 2-stratified graph can be homogeneously embedded in some 2-stratified graph. For a graph G, a 2-stratified graph F of minimum order in which every 2-stratification of G can be homogeneously embedded is called a fence of G and the order of F is called the fencing number fe(G) of G. The fencing numbers of some well-known classes of graphs are determined. It is shown that if G is a vertex-transitive graph of order n that is not a complete graph then fe(G) = 2n.