Rooted product of graphs

File:Graph-rooted-product.svg

The rooted product of graphs.

In mathematical graph theory, the rooted product (or comb product) of a graph Template:Mvar and a rooted graph Template:Mvar is defined as follows: take Template:Abs copies of Template:Mvar, and for every vertex Template:Mvar of Template:Mvar, identify Template:Mvar with the root node of the Template:Mvar-th copy of Template:Mvar.

More formally, assuming that

$\begin{array}{rl}V(G)& =\{g_1,\dots ,g_n\},\\ V(H)& =\{h_1,\dots ,h_m\},\end{array}$

and that the root node of Template:Mvar is h₁, define

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

where

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

and

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

If Template:Mvar 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.

Applications

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.

If Template:Mvar is a two-vertex complete graph K₂, then for any graph Template:Mvar, the rooted product of Template:Mvar and Template:Mvar has domination number exactly half of its number of vertices. Every connected graph in which the domination number is half the number of vertices arises in this way, with the exception of the four-vertex cycle graph. These graphs can be used to generate examples in which the bound of Vizing's conjecture, an unproven inequality between the domination number of the graphs in a different graph product, the cartesian product of graphs, is exactly met . They are also well-covered graphs.

References

• .
• .
• .