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
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
//! Banner-free graph predicate (ALGO-PR-106).
//!
//! A graph is banner-free if it contains no induced banner. The banner
//! (also called flag) is a `C_4` (4-cycle) with one pendant edge:
//! 5 vertices {a, b, c, d, e} with edges {a-b, b-c, c-d, d-a, a-e}
//! and no diagonals or other edges. 5 edges total.
//!
//! For directed graphs, the function returns `false`.
use crate::core::{Graph, IgraphResult};
/// Check whether a graph is banner-free (no induced banner / flag).
///
/// The banner is a `C_4` plus one pendant edge from a cycle vertex.
///
/// Returns `false` for directed graphs.
///
/// # Examples
///
/// ```
/// use rust_igraph::{Graph, is_banner_free};
///
/// // `C_4` is banner-free (no pendant)
/// 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_banner_free(&g).unwrap());
///
/// // Banner: `C_4` {0,1,2,3} + pendant 0-4
/// 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, 0).unwrap();
/// g.add_edge(0, 4).unwrap();
/// assert!(!is_banner_free(&g).unwrap());
/// ```
pub fn is_banner_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 banner: C_4 {a,b,c,d} with edges a-b, b-c, c-d, d-a
// (no diagonals a-c, b-d) plus pendant e adjacent to exactly one
// cycle vertex, say a, and not adjacent to b, c, d.
//
// Strategy: find each induced C_4 (a-b-c-d-a with no diagonals),
// then check if any cycle vertex has a neighbor outside the cycle
// that is not adjacent to the other 3 cycle vertices.
for a in 0..n {
let ai = a as usize;
for &b in &nbrs_list[ai] {
if b <= a {
continue;
}
let bi = b as usize;
for &c in &nbrs_list[bi] {
if c == a {
continue;
}
let ci = c as usize;
if adj[ai][ci] {
continue;
}
// a-b-c path, a and c not adjacent
for &d in &nbrs_list[ci] {
if d == b || d == a {
continue;
}
let di = d as usize;
if !adj[ai][di] || adj[bi][di] {
continue;
}
// Induced C_4: a-b-c-d-a (diags a-c and b-d absent)
// Check each cycle vertex for a pendant
if has_pendant(&adj, &nbrs_list, ai, bi, ci, di) {
return Ok(false);
}
if has_pendant(&adj, &nbrs_list, bi, ai, ci, di) {
return Ok(false);
}
if has_pendant(&adj, &nbrs_list, ci, ai, bi, di) {
return Ok(false);
}
if has_pendant(&adj, &nbrs_list, di, ai, bi, ci) {
return Ok(false);
}
}
}
}
}
Ok(true)
}
/// Check if vertex `v` has a neighbor outside {v, o1, o2, o3} that is
/// not adjacent to o1, o2, or o3.
fn has_pendant(
adj: &[Vec<bool>],
nbrs_list: &[Vec<u32>],
v: usize,
o1: usize,
o2: usize,
o3: usize,
) -> bool {
for &e_u32 in &nbrs_list[v] {
let e = e_u32 as usize;
if e == o1 || e == o2 || e == o3 {
continue;
}
if !adj[o1][e] && !adj[o2][e] && !adj[o3][e] {
return true;
}
}
false
}
#[cfg(test)]
mod tests {
use super::*;
#[test]
fn empty_graph() {
let g = Graph::with_vertices(0);
assert!(is_banner_free(&g).unwrap());
}
#[test]
fn small_graphs() {
let g = Graph::with_vertices(4);
assert!(is_banner_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_banner_free(&g).unwrap());
}
#[test]
fn c4_banner_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();
g.add_edge(3, 0).unwrap();
assert!(is_banner_free(&g).unwrap());
}
#[test]
fn banner() {
// `C_4` {0,1,2,3} + pendant 0-4
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, 0).unwrap();
g.add_edge(0, 4).unwrap();
assert!(!is_banner_free(&g).unwrap());
}
#[test]
fn k5_banner_free() {
// `K_5`: no induced `C_4` (every 4-cycle has a chord)
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_banner_free(&g).unwrap());
}
#[test]
fn path_banner_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_banner_free(&g).unwrap());
}
#[test]
fn star_banner_free() {
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_banner_free(&g).unwrap());
}
#[test]
fn c5_banner_free() {
// `C_5` has no induced `C_4` → banner-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_banner_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_banner_free(&g).unwrap());
}
#[test]
fn pendant_connected_to_non_cycle_vertex() {
// `C_4` + pendant but pendant also connects to another cycle vertex
// → not an induced banner
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, 0).unwrap();
g.add_edge(0, 4).unwrap();
g.add_edge(1, 4).unwrap();
assert!(is_banner_free(&g).unwrap());
}
#[test]
fn cube_not_banner_free() {
// Cube `Q_3` has induced `C_4`. Each `C_4` vertex has a neighbor
// outside the cycle → likely has an induced banner.
let mut g = Graph::with_vertices(8);
g.add_edge(0, 1).unwrap();
g.add_edge(0, 2).unwrap();
g.add_edge(0, 4).unwrap();
g.add_edge(1, 3).unwrap();
g.add_edge(1, 5).unwrap();
g.add_edge(2, 3).unwrap();
g.add_edge(2, 6).unwrap();
g.add_edge(3, 7).unwrap();
g.add_edge(4, 5).unwrap();
g.add_edge(4, 6).unwrap();
g.add_edge(5, 7).unwrap();
g.add_edge(6, 7).unwrap();
assert!(!is_banner_free(&g).unwrap());
}
#[test]
fn k33_banner_free() {
// `K_{3,3}`: has induced `C_4` but every outside vertex connects
// to exactly 2 cycle vertices, never just 1 → banner-free
let mut g = Graph::with_vertices(6);
for i in 0..3u32 {
for j in 3..6u32 {
g.add_edge(i, j).unwrap();
}
}
assert!(is_banner_free(&g).unwrap());
}
#[test]
fn c4_with_isolated_pendant_not_banner_free() {
// `C_4` {0,1,2,3} + pendant 3-4 (4 adj to 3 only)
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, 0).unwrap();
g.add_edge(3, 4).unwrap();
assert!(!is_banner_free(&g).unwrap());
}
}