1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
//! `P_5`-free graph predicate (ALGO-PR-109).
//!
//! A graph is `P_5`-free if it contains no induced path on 5 vertices.
//! `P_5`-free graphs generalize cographs (which are `P_4`-free) and
//! appear in results on well-quasi-ordering and efficient algorithms.
//!
//! For directed graphs, the function returns `false`.
use crate::core::{Graph, IgraphResult};
/// Check whether a graph is `P_5`-free (no induced path on 5 vertices).
///
/// An induced `P_5` is 5 distinct vertices a-b-c-d-e forming a path
/// with edges {a-b, b-c, c-d, d-e} and no other edges among them.
///
/// Returns `false` for directed graphs.
///
/// # Examples
///
/// ```
/// use rust_igraph::{Graph, is_p5_free};
///
/// // `P_4` is `P_5`-free
/// 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_p5_free(&g).unwrap());
///
/// // `P_5` is NOT `P_5`-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();
/// assert!(!is_p5_free(&g).unwrap());
/// ```
pub fn is_p5_free(graph: &Graph) -> IgraphResult<bool> {
if graph.is_directed() {
return Ok(false);
}
let n = graph.vcount();
if n < 5 {
return Ok(true);
}
let n_usize = n as usize;
let mut adj = vec![vec![false; n_usize]; n_usize];
let mut nbrs_list: Vec<Vec<u32>> = Vec::with_capacity(n_usize);
for v in 0..n {
let nbrs = graph.neighbors(v)?;
for &w in &nbrs {
adj[v as usize][w as usize] = true;
}
nbrs_list.push(nbrs);
}
// Induced P_5: a-b-c-d-e with edges a-b, b-c, c-d, d-e and
// no other edges among {a,b,c,d,e}:
// a-c, a-d, a-e, b-d, b-e, c-e all absent.
//
// Strategy: for each edge (b,c), find a adjacent to b but not c,
// then d adjacent to c but not a or b, then e adjacent to d but
// not a, b, or c.
for b in 0..n {
let bi = b as usize;
for &c in &nbrs_list[bi] {
let ci = c as usize;
for &a in &nbrs_list[bi] {
if a == c {
continue;
}
let ai = a as usize;
if adj[ai][ci] {
continue;
}
// a-b edge, a not adj to c
for &d in &nbrs_list[ci] {
if d == b {
continue;
}
let di = d as usize;
if adj[ai][di] || adj[bi][di] {
continue;
}
// c-d edge, d not adj to a or b
for &e in &nbrs_list[di] {
if e == c || e == b || e == a {
continue;
}
let ei = e as usize;
if !adj[ai][ei] && !adj[bi][ei] && !adj[ci][ei] {
return Ok(false);
}
}
}
}
}
}
Ok(true)
}
#[cfg(test)]
mod tests {
use super::*;
#[test]
fn empty_graph() {
let g = Graph::with_vertices(0);
assert!(is_p5_free(&g).unwrap());
}
#[test]
fn small_graphs() {
let g = Graph::with_vertices(4);
assert!(is_p5_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_p5_free(&g).unwrap());
}
#[test]
fn p4_is_p5_free() {
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_p5_free(&g).unwrap());
}
#[test]
fn p5() {
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_p5_free(&g).unwrap());
}
#[test]
fn c5_not_p5_free() {
// `C_5`: 0-1-2-3-4-0. The subpath 0-1-2-3-4 would be `P_5`
// if 0-4 were absent. But 0-4 IS present → induced subgraph
// on {0,1,2,3,4} has edge 0-4. Try {1,2,3,4,0}: edges 1-2,
// 2-3, 3-4, 4-0. Is 1-0 present? No (edge is 0-1, yes it is).
// So we have edges 1-2, 2-3, 3-4, 4-0, 0-1 → `C_5`, not `P_5`.
// Any 5-vertex subset of `C_5` is `C_5` itself. So `C_5` is `P_5`-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_p5_free(&g).unwrap());
}
#[test]
fn c6_not_p5_free() {
// `C_6`: any 5 consecutive vertices form an induced `P_5`
// e.g., {0,1,2,3,4} has edges 0-1, 1-2, 2-3, 3-4 and no
// chord (0-4 absent in `C_6`).
let mut g = Graph::with_vertices(6);
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, 5).unwrap();
g.add_edge(5, 0).unwrap();
assert!(!is_p5_free(&g).unwrap());
}
#[test]
fn k5_p5_free() {
// `K_5`: every pair is adjacent → no induced path of length > 1
let mut g = Graph::with_vertices(5);
for i in 0..5u32 {
for j in (i + 1)..5 {
g.add_edge(i, j).unwrap();
}
}
assert!(is_p5_free(&g).unwrap());
}
#[test]
fn star_p5_free() {
// Star: every path through the center has length 2. No `P_5`.
let mut g = Graph::with_vertices(6);
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(0, 5).unwrap();
assert!(is_p5_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_p5_free(&g).unwrap());
}
#[test]
fn cograph_is_p5_free() {
// `K_2` union `K_3`: {0,1} complete, {2,3,4} complete, no edges between.
// Cographs are `P_4`-free, hence also `P_5`-free.
let mut g = Graph::with_vertices(5);
g.add_edge(0, 1).unwrap();
g.add_edge(2, 3).unwrap();
g.add_edge(3, 4).unwrap();
g.add_edge(2, 4).unwrap();
assert!(is_p5_free(&g).unwrap());
}
#[test]
fn bull_not_p5_free() {
// Bull: triangle {0,1,2} + pendants 1-3, 2-4.
// Path 3-1-0-2-4: edges 3-1, 1-0, 0-2, 2-4. Chords: 1-2 present!
// So {3,1,0,2,4} is not an induced `P_5` (has chord 1-2).
// Try another path: 3-1-2-0-? No pendant from 0. So bull is `P_5`-free.
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_p5_free(&g).unwrap());
}
#[test]
fn petersen_not_p5_free() {
// Petersen graph: diameter 2, every pair at distance ≤ 2.
// But it has induced `P_5` (e.g., any path through two
// non-adjacent vertices on the outer and inner pentagons).
let mut g = Graph::with_vertices(10);
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();
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();
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_p5_free(&g).unwrap());
}
#[test]
fn two_disjoint_edges_p5_free() {
let mut g = Graph::with_vertices(4);
g.add_edge(0, 1).unwrap();
g.add_edge(2, 3).unwrap();
assert!(is_p5_free(&g).unwrap());
}
}