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+# Kruskal's Minimum Spanning Tree (MST) Algorithm
+#
+# Kruskal's algorithm finds a subset of edges that connects all vertices in a graph
+# without any cycles and with the minimum possible total edge weight.
+#
+# It uses the Disjoint Set Union (DSU) or Union-Find data structure to detect cycles efficiently.
+#
+# Time Complexity: O(E log E) — dominated by sorting edges
+# Space Complexity: O(V)
+#
+# Input: graph as a list of edges (each edge has `u`, `v`, and `weight`)
+# Output: List containing MST edges and total minimum cost
+
+# -------------------------
+# Helper: Disjoint Set (Union-Find)
+# -------------------------
+
+make_set <- function(n) {
+  parent <- 1:n
+  rank <- rep(0, n)
+  return(list(parent = parent, rank = rank))
+}
+
+find_set <- function(parent, v) {
+  if (parent[v] != v) {
+    parent[v] <- find_set(parent, parent[v])  # Path compression
+  }
+  return(parent[v])
+}
+
+union_sets <- function(parent, rank, a, b) {
+  a_root <- find_set(parent, a)
+  b_root <- find_set(parent, b)
+  
+  if (a_root != b_root) {
+    if (rank[a_root] < rank[b_root]) {
+      temp <- a_root
+      a_root <- b_root
+      b_root <- temp
+    }
+    parent[b_root] <- a_root
+    if (rank[a_root] == rank[b_root]) {
+      rank[a_root] <- rank[a_root] + 1
+    }
+  }
+  return(list(parent = parent, rank = rank))
+}
+
+# -------------------------
+# Kruskal's MST Algorithm
+# -------------------------
+
+kruskal_mst <- function(edges, num_vertices) {
+  # Sort edges by weight
+  edges <- edges[order(sapply(edges, function(e) e$weight))]
+  
+  # Initialize disjoint sets
+  dsu <- make_set(num_vertices)
+  parent <- dsu$parent
+  rank <- dsu$rank
+  
+  mst_edges <- list()
+  total_cost <- 0
+  
+  # Process each edge
+  for (edge in edges) {
+    u <- edge$u
+    v <- edge$v
+    w <- edge$weight
+    
+    u_root <- find_set(parent, u)
+    v_root <- find_set(parent, v)
+    
+    # If u and v are in different sets, include this edge
+    if (u_root != v_root) {
+      mst_edges <- append(mst_edges, list(edge))
+      total_cost <- total_cost + w
+      merged <- union_sets(parent, rank, u_root, v_root)
+      parent <- merged$parent
+      rank <- merged$rank
+    }
+  }
+  
+  return(list(
+    mst_edges = mst_edges,
+    total_cost = total_cost
+  ))
+}
+
+# -------------------------
+# Example usage and testing
+# -------------------------
+
+cat("=== Kruskal's Minimum Spanning Tree Algorithm ===\n")
+
+# Example undirected weighted graph (edges list)
+# Graph:
+# 1 --(4)-- 2
+# 1 --(3)-- 3
+# 2 --(1)-- 3
+# 2 --(2)-- 4
+# 3 --(5)-- 4
+edges <- list(
+  list(u = 1, v = 2, weight = 4),
+  list(u = 1, v = 3, weight = 3),
+  list(u = 2, v = 3, weight = 1),
+  list(u = 2, v = 4, weight = 2),
+  list(u = 3, v = 4, weight = 5)
+)
+
+cat("Graph edges:\n")
+for (e in edges) {
+  cat(paste0("(", e$u, " -- ", e$v, ") weight = ", e$weight, "\n"))
+}
+
+cat("\nRunning Kruskal's MST:\n")
+result <- kruskal_mst(edges, num_vertices = 4)
+
+cat("MST Edges:\n")
+for (e in result$mst_edges) {
+  cat(paste0("(", e$u, " -- ", e$v, ") weight = ", e$weight, "\n"))
+}
+
+cat("Total Minimum Cost:", result$total_cost, "\n")