## Feb 13, 2010

### Rooted product of graphs - Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Rooted_product_of_graphs

The rooted product of graphs.

In mathematical graph theory, the rooted product of a graph G and a rooted graph H is defined as follows: take |V(G)| copies of H, and for every vertex vi of G, identify vi with the root node of the i-th copy of H.

More formally, assuming that V(G) = {g1, ..., gn}, V(H) = {h1, ..., hm} and that the root node of H is h1, define

$G \circ H := (V, E)$

where

$V = \left\{(g_i, h_j): 1\leq i\leq n, 1\leq j\leq m\right\}$

and

$E = \left\{((g_i, h_1), (g_k, h_1)): (g_i, g_k) \in E(G)\right\} \cup \bigcup_{i=1}^n \left\{((g_i, h_j), (g_i, h_k)): (h_j, h_k) \in E(H)\right\}$

If G is also rooted at g1, one can view the product itself as rooted, at (g1, h1). The rooted product is a subgraph of the cartesian product of the same two graphs.

The rooted product is especially relevant for trees, as the rooted product of two trees is another tree. For instance, Koh et al. (1980) used rooted products to find graceful numberings for a wide family of trees.

## References

Koh, K. M.; Rogers, D. G.; Tan, T. (1980). "Products of graceful trees". Discrete Mathematics 31 (3): 279–292. doi:10.1016/0012-365X(80)90139-9. MR0584121.