Editing Rooted product of graphs

{{short description|Binary operation performed on graphs}}
[[Image:Graph-rooted-product.svg|thumb|upright=1.35|The rooted product of graphs.]]

In mathematical [[graph theory]], the '''rooted product''' of a [[Graph (discrete mathematics)|graph]] {{mvar|G}} and a [[rooted graph]] {{mvar|H}} is defined as follows: take {{math|{{abs|''V''(''G'')}}}} copies of {{mvar|H}}, and for every vertex {{mvar|v{{sub|i}}}} of {{mvar|G}}, identify {{mvar|v{{sub|i}}}} with the root node of the {{mvar|i}}-th copy of {{mvar|H}}.

More formally, assuming that 
:<math>\begin{align}
V(G) &= \{g_1, \ldots, g_n \}, \\
V(H) &= \{h_1, \ldots, h_m \},
\end{align}</math>

and that the root node of {{mvar|H}} is {{math|''h''{{sub|1}}}}, define

:<math>G \circ H := (V, E)</math>,

where

:<math>V = \left\{(g_i, h_j): 1\leq i\leq n, 1\leq j\leq m\right\}</math>

and

:<math>E = \Bigl\{\bigl((g_i, h_1), (g_k, h_1)\bigr): (g_i, g_k) \in E(G)\Bigr\} \cup \bigcup_{i=1}^n \Bigl\{\bigl((g_i, h_j), (g_i, h_k)\bigr): (h_j, h_k) \in E(H)\Bigr\}</math>.

If {{mvar|G}} is also rooted at {{math|''g''<sub>1</sub>}}, one can view the product itself as rooted, at {{math|(''g''<sub>1</sub>, ''h''<sub>1</sub>)}}. The rooted product is a [[Glossary of graph theory#Subgraphs|subgraph]] of the [[cartesian product of graphs|cartesian product]] of the same two graphs.

==Applications==
The rooted product is especially relevant for [[tree (graph theory)|trees]], as the rooted product of two trees is another tree. For instance, Koh et al. (1980) used rooted products to find [[Edge-graceful labeling|graceful numberings]] for a wide family of trees.

If {{mvar|H}} is a two-vertex complete graph {{math|''K''<sub>2</sub>}}, then for any graph {{mvar|G}}, the rooted product of {{mvar|G}} and {{mvar|H}} has [[dominating set|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 {{harv|Fink|Jacobson|Kinch|Roberts|1985}}. They are also [[well-covered graph]]s.

==References==
*{{citation
 | last1 = Godsil | first1 = C. D. | author1-link = Chris Godsil | author2-link = Brendan McKay (mathematician) | last2 = McKay | first2 = B. D.
 | title = A new graph product and its spectrum
 | journal = Bull. Austral. Math. Soc.
 | volume = 18
 | year = 1978
 | issue = 1
 | pages = 21–28
 | mr = 0494910
 | url = http://cs.anu.edu.au/~bdm/papers/RootedProduct.pdf
 | doi = 10.1017/S0004972700007760| doi-access = free
 }}.
*{{citation
 | last1 = Fink | first1 = J. F.
 | last2 = Jacobson | first2 = M. S.
 | last3 = Kinch | first3 = L. F.
 | last4 = Roberts | first4 = J.
 | title = On graphs having domination number half their order
 | year = 1985
 | journal = Period. Math. Hungar.
 | volume = 16
 | pages = 287–293
 | mr = 0833264
 | doi = 10.1007/BF01848079
 | issue = 4}}.
*{{citation
 | last1 = Koh | first1 = K. M. | last2 = Rogers | first2 = D. G. | last3 = Tan | first3 = T.
 | title = Products of graceful trees
 | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
 | volume = 31
 | year = 1980
 | issue = 3
 | pages = 279–292
 | mr = 0584121
 | doi = 10.1016/0012-365X(80)90139-9| doi-access = free
 }}.

[[Category:Graph products]]