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//! Claw-free graph predicate (ALGO-PR-096).
//!
//! A graph is claw-free if it contains no induced subgraph
//! isomorphic to `K_{1,3}` (the complete bipartite graph with parts
//! of size 1 and 3, also called the "claw" or "star on 3 edges").
//!
//! Equivalently, for every vertex v, the open neighborhood N(v) does
//! not contain three mutually non-adjacent vertices (i.e., no
//! independent set of size 3 in N(v)).
//!
//! Claw-free graphs include line graphs and complements of
//! triangle-free graphs.
//!
//! For directed graphs, the function returns `false`.
use crate::core::{Graph, IgraphResult};
/// Check whether a graph is claw-free.
///
/// A claw-free graph has no induced `K_{1,3}`. This is checked by
/// verifying that for every vertex, no three of its neighbors are
/// mutually non-adjacent.
///
/// Returns `false` for directed graphs.
///
/// # Examples
///
/// ```
/// use rust_igraph::{Graph, is_claw_free};
///
/// // Triangle is claw-free
/// 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_claw_free(&g).unwrap());
///
/// // Star `K_{1,3}` IS a claw
/// 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();
/// assert!(!is_claw_free(&g).unwrap());
/// ```
pub fn is_claw_free(graph: &Graph) -> IgraphResult<bool> {
if graph.is_directed() {
return Ok(false);
}
let n = graph.vcount();
if n <= 3 {
// Can't have K_{1,3} with fewer than 4 vertices
return Ok(true);
}
// Build adjacency for fast pair lookup
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;
}
}
// For each vertex v, check if N(v) contains an independent set of size 3
for v in 0..n {
let nbrs = graph.neighbors(v)?;
let deg = nbrs.len();
if deg < 3 {
continue;
}
// Check all triples of neighbors
for i in 0..deg {
for j in (i + 1)..deg {
if adj[nbrs[i] as usize][nbrs[j] as usize] {
continue;
}
// nbrs[i] and nbrs[j] are non-adjacent
for k in (j + 1)..deg {
if !adj[nbrs[i] as usize][nbrs[k] as usize]
&& !adj[nbrs[j] as usize][nbrs[k] as usize]
{
// Found independent triple in N(v) → induced K_{1,3}
return Ok(false);
}
}
}
}
}
Ok(true)
}
#[cfg(test)]
mod tests {
use super::*;
#[test]
fn empty_graph() {
let g = Graph::with_vertices(0);
assert!(is_claw_free(&g).unwrap());
}
#[test]
fn single_vertex() {
let g = Graph::with_vertices(1);
assert!(is_claw_free(&g).unwrap());
}
#[test]
fn single_edge() {
let mut g = Graph::with_vertices(2);
g.add_edge(0, 1).unwrap();
assert!(is_claw_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_claw_free(&g).unwrap());
}
#[test]
fn k4() {
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_claw_free(&g).unwrap());
}
#[test]
fn star_k13_is_claw() {
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();
assert!(!is_claw_free(&g).unwrap());
}
#[test]
fn star_k14_not_claw_free() {
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();
assert!(!is_claw_free(&g).unwrap());
}
#[test]
fn path_claw_free() {
// Path 0-1-2-3: max degree 2, can't have K_{1,3}
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();
assert!(is_claw_free(&g).unwrap());
}
#[test]
fn cycle_c5_claw_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_claw_free(&g).unwrap());
}
#[test]
fn line_graph_of_k4() {
// Line graph of K4 has 6 vertices (one per edge of K4)
// Line graphs are always claw-free (Whitney's theorem)
// Edges of K4: 01, 02, 03, 12, 13, 23
// Two edges adjacent iff they share a vertex
let mut g = Graph::with_vertices(6);
// 01-02, 01-03, 01-12, 01-13
g.add_edge(0, 1).unwrap(); // 01-02
g.add_edge(0, 2).unwrap(); // 01-03
g.add_edge(0, 3).unwrap(); // 01-12
g.add_edge(0, 4).unwrap(); // 01-13
// 02-03, 02-12, 02-23
g.add_edge(1, 2).unwrap(); // 02-03
g.add_edge(1, 3).unwrap(); // 02-12
g.add_edge(1, 5).unwrap(); // 02-23
// 03-13, 03-23
g.add_edge(2, 4).unwrap(); // 03-13
g.add_edge(2, 5).unwrap(); // 03-23
// 12-13, 12-23
g.add_edge(3, 4).unwrap(); // 12-13
g.add_edge(3, 5).unwrap(); // 12-23
// 13-23
g.add_edge(4, 5).unwrap(); // 13-23
assert!(is_claw_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_claw_free(&g).unwrap());
}
#[test]
fn diamond_claw_free() {
// Diamond: K4 minus one edge
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(2, 3).unwrap();
// Vertex 0 has neighbors {1, 2, 3}; 1-2 adjacent, 2-3 adjacent,
// but 1-3 not adjacent. That's only 2 non-adjacent, need 3.
// No vertex has 3 mutually non-adjacent neighbors.
assert!(is_claw_free(&g).unwrap());
}
#[test]
fn petersen_not_claw_free() {
// Petersen graph is NOT claw-free (3-regular, 10 vertices)
let mut g = Graph::with_vertices(10);
// Outer cycle
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();
// Inner pentagram
g.add_edge(5, 7).unwrap();
g.add_edge(7, 9).unwrap();
g.add_edge(9, 6).unwrap();
g.add_edge(6, 8).unwrap();
g.add_edge(8, 5).unwrap();
// Spokes
g.add_edge(0, 5).unwrap();
g.add_edge(1, 6).unwrap();
g.add_edge(2, 7).unwrap();
g.add_edge(3, 8).unwrap();
g.add_edge(4, 9).unwrap();
assert!(!is_claw_free(&g).unwrap());
}
#[test]
fn two_triangles_sharing_edge_claw_free() {
// 0-1-2-0, 1-2-3-1
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(1, 3).unwrap();
g.add_edge(2, 3).unwrap();
assert!(is_claw_free(&g).unwrap());
}
}