diff --git a/DESCRIPTION b/DESCRIPTION
index 3ce53cc..7b244a4 100644
--- a/DESCRIPTION
+++ b/DESCRIPTION
@@ -1,6 +1,6 @@
 Package: keedwell
 Title: Latin Squares in R
-Version: 0.1.1
+Version: 0.1.1.9000
 Authors@R: 
     person("Matthew", "Henderson", , "matthew.james.henderson@gmail.com", role = c("aut", "cre"),
            comment = c(ORCID = "0000-0001-7949-8208"))
diff --git a/NAMESPACE b/NAMESPACE
index 2ed5ff9..35344d5 100644
--- a/NAMESPACE
+++ b/NAMESPACE
@@ -1,7 +1,13 @@
 # Generated by roxygen2: do not edit by hand
 
+export(add_cols)
 export(add_rows)
 export(create_latin_square)
 export(edge_tbl)
+export(next_col_matching)
+export(next_col_random)
 export(next_row)
+export(next_row_matching)
+export(next_row_random)
 export(to_tidygraph)
+export(to_tidygraph_2)
diff --git a/R/add-cols.R b/R/add-cols.R
new file mode 100644
index 0000000..bbcde27
--- /dev/null
+++ b/R/add-cols.R
@@ -0,0 +1,20 @@
+#' Add new columns to a latin rectangle
+#'
+#' @param R A latin rectangle
+#' @param cols Indices of columns to add
+#' @param l_order Dimension of latin square
+#' @param strategy Strategy for filling columns
+#'
+#' @return A latin rectangle
+#' @export
+add_cols <- function(R, cols, l_order, strategy = next_col_matching) {
+  
+  for (i in cols) {
+    
+    R <- strategy(R, i, l_order)
+    
+  }
+  
+  return(R)
+  
+}
diff --git a/R/edge-tbl.R b/R/edge-tbl.R
index 6e76887..fea2042 100644
--- a/R/edge-tbl.R
+++ b/R/edge-tbl.R
@@ -29,5 +29,31 @@ edge_tbl <- function(R, i, l_order = 3) {
   ) |>
     dplyr::mutate(from = paste0("c", i))
   
+  return(edge_df)
+}
+
+#' Symbols missing from rows data frame
+#'
+#' @param R latin rectangle
+#' @param i row index
+#' @param l_order size of latin square R is going to be embedded into
+#'
+#' @return The edge data frame.
+edge_tbl_2 <- function(R, i, l_order = 3) {
+  
+  all_symbols <- 1:l_order
+  
+  # symbols used in row i
+  used <- R |> dplyr::filter(row == i) |> dplyr::pull(symbol)
+  
+  # symbols missing from row i
+  missing <- setdiff(all_symbols, used)
+  
+  # edge data frame for column i
+  edge_df <- tibble::tibble(
+    to = paste0("s", missing)
+  ) |>
+    dplyr::mutate(from = paste0("r", i))
+  
   return(edge_df)
 }
\ No newline at end of file
diff --git a/R/next-col.R b/R/next-col.R
new file mode 100644
index 0000000..cc337e3
--- /dev/null
+++ b/R/next-col.R
@@ -0,0 +1,69 @@
+#' Matching strategy for adding new columns
+#'
+#' @param R a latin rectangle
+#' @param i column index
+#' @param l_order dimension
+#'
+#' @return a latin rectangle with more columns
+#' @export
+next_col_matching <- function(R, i, l_order) {
+  
+  n_rows <- max(R$row)
+  n_cols <- max(R$column)
+  
+  bg <- to_tidygraph_2(R, l_order, n_rows, n_cols)
+  
+  m <- igraph::max_bipartite_match(bg)
+  
+  # names of edges in the matching
+  matching_names <- match(m$matching, names(m$matching))
+  
+  # add a matching indicator to the edges
+  bg <- bg |>
+    tidygraph::activate(edges) |>
+    dplyr::mutate(
+      matching = to == matching_names[from]
+    ) |>
+    dplyr::filter(from <= n_rows)
+  
+  # just the matching itself, as a graph
+  mg <- bg |>
+    tidygraph::activate(edges) |>
+    dplyr::filter(matching)
+  
+  EE <- igraph::ends(mg, igraph::E(mg))
+  
+  R |>
+    dplyr::bind_rows(
+      tibble::tibble(
+        column = rep(i, n_rows),
+        row = 1:n_rows,
+        symbol = as.numeric(gsub("s", "", EE[, 2]))
+      )
+    )
+  
+}
+
+#' Random strategy for choosing a new columns
+#'
+#' @param R a latin rectangle
+#' @param i column index
+#' @param l_order dimension
+#'
+#' @return A latin rectangle with more columns
+#' @export
+next_col_random <- function(R, i, l_order) {
+  
+  n_rows <- max(R$row)
+  n_cols <- max(R$column)
+  
+  R |>
+    dplyr::bind_rows(
+      tibble::tibble(
+        column = rep(i, n_rows),
+        row = 1:n_rows,
+        symbol = sample(1:l_order, n_rows)
+      )
+    )
+  
+}
\ No newline at end of file
diff --git a/R/next-row.R b/R/next-row.R
index 09b989b..15bcba0 100644
--- a/R/next-row.R
+++ b/R/next-row.R
@@ -36,4 +36,74 @@ next_row <- function(R, l_order) {
   
   EE <- igraph::ends(mg, igraph::E(mg))
   return(as.numeric(gsub("s", "", EE[,2])))
-}
\ No newline at end of file
+}
+
+#' Find a random new row for a latin rectangle
+#'
+#' @param R A latin rectangle
+#' @param i Number of columns to add
+#' @param l_order Order of R
+#'
+#' @return A latin square with more rows.
+#' @export
+next_row_random <- function(R, i, l_order) {
+  
+  R |>
+    dplyr::bind_rows(
+      tibble::tibble(
+        row = rep(i, l_order),
+        column = 1:l_order,
+        symbol = sample(1:l_order, l_order)
+      )
+    )
+  
+}
+
+#' Find a compatible row for extending a latin rectangle
+#'
+#' Given an input latin rectangle this function will
+#' generate a new row that can be added to the latin
+#' rectangle.
+#'
+#' The method used is to create a bipartite graph
+#' with vertex partitions for columns and symbols
+#' missing from columns and then find a maximum
+#' matching in that bipartite graph.
+#'
+#' @param R A latin rectangle
+#' @param i Number of columns to add
+#' @param l_order Order of R
+#'
+#' @return A latin rectangle with more rows.
+#' @export
+next_row_matching <- function(R, i, l_order) {
+  bg <- to_tidygraph(R, l_order)
+  
+  m <- igraph::max_bipartite_match(bg)
+  
+  # names of edges in the matching
+  matching_names <- match(m$matching, names(m$matching))
+  
+  # add a matching indicator to the edges
+  bg <- bg |>
+    tidygraph::activate(edges) |>
+    dplyr::mutate(
+      matching = to == matching_names[from]
+    )
+  
+  # just the matching itself, as a graph
+  mg <- bg |>
+    tidygraph::activate(edges) |>
+    dplyr::filter(matching)
+  
+  EE <- igraph::ends(mg, igraph::E(mg))
+  
+  R |>
+    dplyr::bind_rows(
+      tibble::tibble(
+        row = rep(i, l_order),
+        column = 1:l_order,
+        symbol = as.numeric(gsub("s", "", EE[,2]))
+      )
+    )
+}
diff --git a/R/to-tidygraph.R b/R/to-tidygraph.R
index 62bfbc3..543a49d 100644
--- a/R/to-tidygraph.R
+++ b/R/to-tidygraph.R
@@ -29,4 +29,44 @@ to_tidygraph <- function(R, l_order = 3) {
   
   tidygraph::tbl_graph(nodes = l_nodes, edges = l_edges)
 
-}
\ No newline at end of file
+}
+
+#' Symbols missing from rows bipartite graph
+#'
+
+#' @param R A latin rectangle.
+#' @param l_order Order of R.
+#' @param n_rows Number of rows.
+#' @param n_cols Number of columns.
+#'
+#' @return A bipartite graph.
+#' @export
+to_tidygraph_2 <- function(R, l_order, n_rows, n_cols) {
+  
+  ## VERTEX DATA FRAME
+  row_vertices <- paste0("r", 1:n_rows)
+  symbol_vertices <- paste0("s", 1:l_order)
+  
+  n_dummy_nodes <- l_order - n_rows
+  dummy_vertices <- paste0("d", 1:n_dummy_nodes)
+  
+  l_nodes <- tibble::tibble(
+    name = c(row_vertices, dummy_vertices, symbol_vertices),
+    type = c(rep(TRUE, n_rows + n_dummy_nodes), rep(FALSE, l_order))
+  )
+  
+  ## EDGE DATA FRAME
+  f <- function(i) return(edge_tbl_2(R, i, l_order))
+  
+  l_edges <- purrr::map_df(1:n_rows, f)
+  
+  G <- tidygraph::tbl_graph(nodes = l_nodes, edges = l_edges)
+  
+  d_edges <- tibble::tibble(
+    to = rep(symbol_vertices, as.numeric(l_order - n_cols - igraph::degree(G, symbol_vertices))),
+    from = rep(dummy_vertices, each = l_order - n_cols)
+  )
+  
+  tidygraph::tbl_graph(nodes = l_nodes, edges = dplyr::bind_rows(l_edges, d_edges))
+  
+}
diff --git a/man/add_cols.Rd b/man/add_cols.Rd
new file mode 100644
index 0000000..d949a73
--- /dev/null
+++ b/man/add_cols.Rd
@@ -0,0 +1,23 @@
+% Generated by roxygen2: do not edit by hand
+% Please edit documentation in R/add-cols.R
+\name{add_cols}
+\alias{add_cols}
+\title{Add new columns to a latin rectangle}
+\usage{
+add_cols(R, cols, l_order, strategy = next_col_matching)
+}
+\arguments{
+\item{R}{A latin rectangle}
+
+\item{cols}{Indices of columns to add}
+
+\item{l_order}{Dimension of latin square}
+
+\item{strategy}{Strategy for filling columns}
+}
+\value{
+A latin rectangle
+}
+\description{
+Add new columns to a latin rectangle
+}
diff --git a/man/edge_tbl_2.Rd b/man/edge_tbl_2.Rd
new file mode 100644
index 0000000..e5082e5
--- /dev/null
+++ b/man/edge_tbl_2.Rd
@@ -0,0 +1,21 @@
+% Generated by roxygen2: do not edit by hand
+% Please edit documentation in R/edge-tbl.R
+\name{edge_tbl_2}
+\alias{edge_tbl_2}
+\title{Symbols missing from rows data frame}
+\usage{
+edge_tbl_2(R, i, l_order = 3)
+}
+\arguments{
+\item{R}{latin rectangle}
+
+\item{i}{row index}
+
+\item{l_order}{size of latin square R is going to be embedded into}
+}
+\value{
+The edge data frame.
+}
+\description{
+Symbols missing from rows data frame
+}
diff --git a/man/next_col_matching.Rd b/man/next_col_matching.Rd
new file mode 100644
index 0000000..b6f866c
--- /dev/null
+++ b/man/next_col_matching.Rd
@@ -0,0 +1,21 @@
+% Generated by roxygen2: do not edit by hand
+% Please edit documentation in R/next-col.R
+\name{next_col_matching}
+\alias{next_col_matching}
+\title{Matching strategy for adding new columns}
+\usage{
+next_col_matching(R, i, l_order)
+}
+\arguments{
+\item{R}{a latin rectangle}
+
+\item{i}{column index}
+
+\item{l_order}{dimension}
+}
+\value{
+a latin rectangle with more columns
+}
+\description{
+Matching strategy for adding new columns
+}
diff --git a/man/next_col_random.Rd b/man/next_col_random.Rd
new file mode 100644
index 0000000..7ba64a1
--- /dev/null
+++ b/man/next_col_random.Rd
@@ -0,0 +1,21 @@
+% Generated by roxygen2: do not edit by hand
+% Please edit documentation in R/next-col.R
+\name{next_col_random}
+\alias{next_col_random}
+\title{Random strategy for choosing a new columns}
+\usage{
+next_col_random(R, i, l_order)
+}
+\arguments{
+\item{R}{a latin rectangle}
+
+\item{i}{column index}
+
+\item{l_order}{dimension}
+}
+\value{
+A latin rectangle with more columns
+}
+\description{
+Random strategy for choosing a new columns
+}
diff --git a/man/next_row_matching.Rd b/man/next_row_matching.Rd
new file mode 100644
index 0000000..78d211e
--- /dev/null
+++ b/man/next_row_matching.Rd
@@ -0,0 +1,29 @@
+% Generated by roxygen2: do not edit by hand
+% Please edit documentation in R/next-row.R
+\name{next_row_matching}
+\alias{next_row_matching}
+\title{Find a compatible row for extending a latin rectangle}
+\usage{
+next_row_matching(R, i, l_order)
+}
+\arguments{
+\item{R}{A latin rectangle}
+
+\item{i}{Number of columns to add}
+
+\item{l_order}{Order of R}
+}
+\value{
+A latin rectangle with more rows.
+}
+\description{
+Given an input latin rectangle this function will
+generate a new row that can be added to the latin
+rectangle.
+}
+\details{
+The method used is to create a bipartite graph
+with vertex partitions for columns and symbols
+missing from columns and then find a maximum
+matching in that bipartite graph.
+}
diff --git a/man/next_row_random.Rd b/man/next_row_random.Rd
new file mode 100644
index 0000000..f959c2c
--- /dev/null
+++ b/man/next_row_random.Rd
@@ -0,0 +1,21 @@
+% Generated by roxygen2: do not edit by hand
+% Please edit documentation in R/next-row.R
+\name{next_row_random}
+\alias{next_row_random}
+\title{Find a random new row for a latin rectangle}
+\usage{
+next_row_random(R, i, l_order)
+}
+\arguments{
+\item{R}{A latin rectangle}
+
+\item{i}{Number of columns to add}
+
+\item{l_order}{Order of R}
+}
+\value{
+A latin square with more rows.
+}
+\description{
+Find a random new row for a latin rectangle
+}
diff --git a/man/to_tidygraph_2.Rd b/man/to_tidygraph_2.Rd
new file mode 100644
index 0000000..d0e4e35
--- /dev/null
+++ b/man/to_tidygraph_2.Rd
@@ -0,0 +1,23 @@
+% Generated by roxygen2: do not edit by hand
+% Please edit documentation in R/to-tidygraph.R
+\name{to_tidygraph_2}
+\alias{to_tidygraph_2}
+\title{Symbols missing from rows bipartite graph}
+\usage{
+to_tidygraph_2(R, l_order, n_rows, n_cols)
+}
+\arguments{
+\item{R}{A latin rectangle.}
+
+\item{l_order}{Order of R.}
+
+\item{n_rows}{Number of rows.}
+
+\item{n_cols}{Number of columns.}
+}
+\value{
+A bipartite graph.
+}
+\description{
+Symbols missing from rows bipartite graph
+}