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//! Paw-free graph predicate (ALGO-PR-099).
//!
//! A graph is paw-free if it contains no induced subgraph isomorphic
//! to the paw graph. The paw is a triangle with one pendant edge
//! (4 vertices, 4 edges): vertices {a, b, c, d} with edges
//! {ab, ac, bc, cd} (d is pendant, adjacent only to c).
//!
//! Paw-free graphs are exactly the graphs where every connected
//! component is either triangle-free or complete.
//!
//! For directed graphs, the function returns `false`.
use crate::core::{Graph, IgraphResult};
/// Check whether a graph is paw-free.
///
/// The paw is a triangle plus a pendant edge. A paw-free graph has
/// no induced paw. Equivalently, every connected component is
/// either triangle-free or complete.
///
/// Returns `false` for directed graphs.
///
/// # Examples
///
/// ```
/// use rust_igraph::{Graph, is_paw_free};
///
/// // Triangle is paw-free (no pendant vertex)
/// 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_paw_free(&g).unwrap());
///
/// // Triangle + pendant: paw!
/// let mut g = Graph::with_vertices(4);
/// g.add_edge(0, 1).unwrap();
/// g.add_edge(1, 2).unwrap();
/// g.add_edge(2, 0).unwrap();
/// g.add_edge(2, 3).unwrap();
/// assert!(!is_paw_free(&g).unwrap());
/// ```
pub fn is_paw_free(graph: &Graph) -> IgraphResult<bool> {
if graph.is_directed() {
return Ok(false);
}
let n = graph.vcount();
if n < 4 {
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;
}
}
// Check for induced paw: find a triangle {a, b, c} where some
// vertex d is adjacent to exactly one of {a, b, c}.
// Equivalently: for each edge (a, b), find common neighbor c
// (forming a triangle), then check if any neighbor of a, b, or c
// is adjacent to exactly one member of the triangle.
for a in 0..n {
let ai = a as usize;
for b in (a + 1)..n {
let bi = b as usize;
if !adj[ai][bi] {
continue;
}
// Find common neighbors of a and b (forming triangles)
for c in (b + 1)..n {
let ci = c as usize;
if !adj[ai][ci] || !adj[bi][ci] {
continue;
}
// Triangle {a, b, c} found. Check if any vertex d
// is adjacent to exactly one of {a, b, c}.
for d in 0..n {
if d == a || d == b || d == c {
continue;
}
let di = d as usize;
let count =
u32::from(adj[ai][di]) + u32::from(adj[bi][di]) + u32::from(adj[ci][di]);
if count == 1 {
return Ok(false);
}
}
}
}
}
Ok(true)
}
#[cfg(test)]
mod tests {
use super::*;
#[test]
fn empty_graph() {
let g = Graph::with_vertices(0);
assert!(is_paw_free(&g).unwrap());
}
#[test]
fn single_vertex() {
let g = Graph::with_vertices(1);
assert!(is_paw_free(&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_paw_free(&g).unwrap());
}
#[test]
fn paw() {
// Triangle 0-1-2 + pendant 2-3
let mut g = Graph::with_vertices(4);
g.add_edge(0, 1).unwrap();
g.add_edge(1, 2).unwrap();
g.add_edge(2, 0).unwrap();
g.add_edge(2, 3).unwrap();
assert!(!is_paw_free(&g).unwrap());
}
#[test]
fn k4_paw_free() {
// K_4 is complete → paw-free (no vertex adjacent to exactly 1 of a triangle)
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();
g.add_edge(2, 3).unwrap();
assert!(is_paw_free(&g).unwrap());
}
#[test]
fn path_paw_free() {
// Path: no triangles → no paw
let mut g = Graph::with_vertices(5);
g.add_edge(0, 1).unwrap();
g.add_edge(1, 2).unwrap();
g.add_edge(2, 3).unwrap();
g.add_edge(3, 4).unwrap();
assert!(is_paw_free(&g).unwrap());
}
#[test]
fn c5_paw_free() {
// C_5 is triangle-free → paw-free
let mut g = Graph::with_vertices(5);
g.add_edge(0, 1).unwrap();
g.add_edge(1, 2).unwrap();
g.add_edge(2, 3).unwrap();
g.add_edge(3, 4).unwrap();
g.add_edge(4, 0).unwrap();
assert!(is_paw_free(&g).unwrap());
}
#[test]
fn diamond_not_paw_free() {
// Diamond: K_4 minus edge 2-3. Triangle {0,1,2} with vertex 3
// adjacent to 0 and 1 but not 2 → vertex 3 adj to 2 of triangle
// Actually: diamond = {01, 02, 03, 12, 13}. Triangle {0,1,2}.
// Vertex 3 is adjacent to 0 and 1 (count=2) → not exactly 1.
// Triangle {0,1,3}: vertex 2 adjacent to 0 and 1 (count=2) → not 1.
// So diamond might actually be paw-free? Let me check:
// Triangle {0,1,2}: vertex 3 adj to 0(yes) 1(yes) 2(no) → count=2
// Triangle {0,1,3}: vertex 2 adj to 0(yes) 1(yes) 3(no) → count=2
// No triangle has a vertex adjacent to exactly 1 member → paw-free!
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_paw_free(&g).unwrap());
}
#[test]
fn bull_not_paw_free() {
// Bull: triangle 0-1-2, pendants 1-3 and 2-4
// Triangle {0,1,2}, vertex 3 adjacent to 1 only → count=1 → paw!
let mut g = Graph::with_vertices(5);
g.add_edge(0, 1).unwrap();
g.add_edge(0, 2).unwrap();
g.add_edge(1, 2).unwrap();
g.add_edge(1, 3).unwrap();
g.add_edge(2, 4).unwrap();
assert!(!is_paw_free(&g).unwrap());
}
#[test]
fn disconnected_triangle_plus_edge() {
// Triangle + isolated edge: no paw (the edge vertex is in
// a different component from the triangle)
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(3, 4).unwrap();
// Triangle {0,1,2}: vertex 3 adj to 0(no) 1(no) 2(no) → count=0
// vertex 4 adj to 0(no) 1(no) 2(no) → count=0
// No paw!
assert!(is_paw_free(&g).unwrap());
}
#[test]
fn directed_returns_false() {
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_paw_free(&g).unwrap());
}
#[test]
fn isolated_vertices() {
let g = Graph::with_vertices(10);
assert!(is_paw_free(&g).unwrap());
}
#[test]
fn windmill_not_paw_free() {
// Wd(3,2): two triangles sharing vertex 0
// 0-1-2-0, 0-3-4-0
// Triangle {0,1,2}: vertex 3 adj to 0 only → count=1 → paw!
let mut g = Graph::with_vertices(5);
g.add_edge(0, 1).unwrap();
g.add_edge(0, 2).unwrap();
g.add_edge(1, 2).unwrap();
g.add_edge(0, 3).unwrap();
g.add_edge(0, 4).unwrap();
g.add_edge(3, 4).unwrap();
assert!(!is_paw_free(&g).unwrap());
}
}