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//! Apex tree predicate (ALGO-PR-128).
//!
//! A graph is an apex tree if there exists a vertex whose removal
//! makes the remaining graph a tree. This is stricter than apex
//! forest (which requires only a forest after removal).
//!
//! Every tree is trivially an apex tree (remove any leaf).
//! Every graph with exactly one cycle where the cycle passes
//! through some vertex is an apex tree.
//!
//! Directed graphs are treated as undirected.
use crate::algorithms::paths::dijkstra::DijkstraMode;
use crate::algorithms::properties::is_tree::is_tree;
use crate::core::{Graph, IgraphResult};
/// Check whether a graph is an apex tree.
///
/// A graph is an apex tree if removing some single vertex yields
/// a tree. Tests each vertex as a candidate apex.
///
/// Directed graphs are treated as undirected.
///
/// # Examples
///
/// ```
/// use rust_igraph::{Graph, is_apex_tree};
///
/// // Triangle: remove any vertex → single edge (tree) ✓
/// let mut g = Graph::with_vertices(3);
/// g.add_edge(0, 1).unwrap();
/// g.add_edge(1, 2).unwrap();
/// g.add_edge(2, 0).unwrap();
/// assert!(is_apex_tree(&g).unwrap());
///
/// // Two disjoint triangles: no single vertex removal yields tree
/// let mut g = Graph::with_vertices(6);
/// g.add_edge(0, 1).unwrap();
/// g.add_edge(1, 2).unwrap();
/// g.add_edge(2, 0).unwrap();
/// g.add_edge(3, 4).unwrap();
/// g.add_edge(4, 5).unwrap();
/// g.add_edge(5, 3).unwrap();
/// assert!(!is_apex_tree(&g).unwrap());
/// ```
pub fn is_apex_tree(graph: &Graph) -> IgraphResult<bool> {
let n = graph.vcount();
if n <= 1 {
return Ok(true);
}
if is_tree(graph, DijkstraMode::All)?.is_some() {
return Ok(true);
}
let n_usize = n as usize;
let mut adj = vec![vec![false; n_usize]; n_usize];
for v in 0..n {
let nbrs = graph.neighbors(v)?;
for &w in &nbrs {
adj[v as usize][w as usize] = true;
adj[w as usize][v as usize] = true;
}
}
for apex in 0..n_usize {
if is_tree_without(&adj, apex, n_usize) {
return Ok(true);
}
}
Ok(false)
}
fn is_tree_without(adj: &[Vec<bool>], removed: usize, n: usize) -> bool {
let remaining = n - 1;
if remaining == 0 {
return true;
}
let mut visited = vec![false; n];
visited[removed] = true;
let start = (0..n).find(|&v| v != removed).unwrap();
let mut stack = vec![start];
visited[start] = true;
let mut visited_count = 1usize;
let mut edge_count = 0usize;
while let Some(u) = stack.pop() {
for (v, &is_adj) in adj[u].iter().enumerate() {
if v == removed || !is_adj {
continue;
}
edge_count += 1;
if !visited[v] {
visited[v] = true;
visited_count += 1;
stack.push(v);
}
}
}
edge_count /= 2;
// Tree iff connected (visited_count == remaining) and edges == remaining - 1
visited_count == remaining && edge_count == remaining - 1
}
#[cfg(test)]
mod tests {
use super::*;
#[test]
fn empty_graph() {
let g = Graph::with_vertices(0);
assert!(is_apex_tree(&g).unwrap());
}
#[test]
fn single_vertex() {
let g = Graph::with_vertices(1);
assert!(is_apex_tree(&g).unwrap());
}
#[test]
fn single_edge() {
let mut g = Graph::with_vertices(2);
g.add_edge(0, 1).unwrap();
assert!(is_apex_tree(&g).unwrap());
}
#[test]
fn triangle() {
let mut g = Graph::with_vertices(3);
g.add_edge(0, 1).unwrap();
g.add_edge(1, 2).unwrap();
g.add_edge(2, 0).unwrap();
assert!(is_apex_tree(&g).unwrap());
}
#[test]
fn tree() {
let mut g = Graph::with_vertices(5);
g.add_edge(0, 1).unwrap();
g.add_edge(1, 2).unwrap();
g.add_edge(1, 3).unwrap();
g.add_edge(3, 4).unwrap();
assert!(is_apex_tree(&g).unwrap());
}
#[test]
fn c4() {
// C_4: remove any vertex → P_3 (tree) ✓
let mut g = Graph::with_vertices(4);
g.add_edge(0, 1).unwrap();
g.add_edge(1, 2).unwrap();
g.add_edge(2, 3).unwrap();
g.add_edge(3, 0).unwrap();
assert!(is_apex_tree(&g).unwrap());
}
#[test]
fn k4() {
// K_4: remove any vertex → K_3 (triangle, NOT a tree).
let mut g = Graph::with_vertices(4);
for i in 0..4u32 {
for j in (i + 1)..4 {
g.add_edge(i, j).unwrap();
}
}
assert!(!is_apex_tree(&g).unwrap());
}
#[test]
fn two_triangles_not_apex_tree() {
let mut g = Graph::with_vertices(6);
g.add_edge(0, 1).unwrap();
g.add_edge(1, 2).unwrap();
g.add_edge(2, 0).unwrap();
g.add_edge(3, 4).unwrap();
g.add_edge(4, 5).unwrap();
g.add_edge(5, 3).unwrap();
assert!(!is_apex_tree(&g).unwrap());
}
#[test]
fn two_triangles_shared_vertex() {
// 0-1-2-0 and 0-3-4-0. Remove 0 → {1,2,3,4} with edges 1-2, 3-4.
// That's a forest, NOT a tree (two components).
// So NOT apex tree (but IS apex forest).
let mut g = Graph::with_vertices(5);
g.add_edge(0, 1).unwrap();
g.add_edge(1, 2).unwrap();
g.add_edge(2, 0).unwrap();
g.add_edge(0, 3).unwrap();
g.add_edge(3, 4).unwrap();
g.add_edge(4, 0).unwrap();
assert!(!is_apex_tree(&g).unwrap());
}
#[test]
fn diamond() {
// Diamond: 0-1,0-2,0-3,1-2,1-3 (K_4 minus 2-3).
// Remove 0: {1,2,3} with edges 1-2,1-3 → star, which is a tree ✓
let mut g = Graph::with_vertices(4);
g.add_edge(0, 1).unwrap();
g.add_edge(0, 2).unwrap();
g.add_edge(0, 3).unwrap();
g.add_edge(1, 2).unwrap();
g.add_edge(1, 3).unwrap();
assert!(is_apex_tree(&g).unwrap());
}
#[test]
fn theta_graph() {
// Theta: 0-1-3 and 0-2-3 (two paths between 0 and 3).
// Remove 0 → {1,2,3} with edges 1-3,2-3 → star on 3 → tree ✓
let mut g = Graph::with_vertices(4);
g.add_edge(0, 1).unwrap();
g.add_edge(1, 3).unwrap();
g.add_edge(0, 2).unwrap();
g.add_edge(2, 3).unwrap();
assert!(is_apex_tree(&g).unwrap());
}
#[test]
fn edgeless_not_apex_tree() {
// 4 isolated vertices: removing any one leaves 3 isolated vertices (forest, not tree)
let g = Graph::with_vertices(4);
assert!(!is_apex_tree(&g).unwrap());
}
#[test]
fn two_isolated() {
// 2 isolated: removing one yields single vertex (tree) ✓
let g = Graph::with_vertices(2);
assert!(is_apex_tree(&g).unwrap());
}
#[test]
fn directed_treated_as_undirected() {
let mut g = Graph::new(3, true).unwrap();
g.add_edge(0, 1).unwrap();
g.add_edge(1, 2).unwrap();
g.add_edge(2, 0).unwrap();
assert!(is_apex_tree(&g).unwrap());
}
#[test]
fn wheel_w4() {
// W_4: center 0, rim 1-2-3-4-1. Remove 0 → C_4 (not a tree).
// Remove rim vertex 1 → 0 connected to 2,3,4; plus 2-3,3-4.
// Edges: 0-2,0-3,0-4,2-3,3-4. 4 vertices, 5 edges. Not tree (4 verts need 3 edges).
// NOT apex tree.
let mut g = Graph::with_vertices(5);
g.add_edge(0, 1).unwrap();
g.add_edge(0, 2).unwrap();
g.add_edge(0, 3).unwrap();
g.add_edge(0, 4).unwrap();
g.add_edge(1, 2).unwrap();
g.add_edge(2, 3).unwrap();
g.add_edge(3, 4).unwrap();
g.add_edge(4, 1).unwrap();
assert!(!is_apex_tree(&g).unwrap());
}
#[test]
fn tree_with_extra_edge() {
// Tree 0-1-2-3 plus edge 0-2. Only one cycle through 0-1-2.
// Remove 0: {1,2,3} with edges 1-2,2-3 → path → tree ✓
let mut g = Graph::with_vertices(4);
g.add_edge(0, 1).unwrap();
g.add_edge(1, 2).unwrap();
g.add_edge(2, 3).unwrap();
g.add_edge(0, 2).unwrap();
assert!(is_apex_tree(&g).unwrap());
}
}