Compare current row value to previous row values

Query duplicated() and change astype to int

df['isDup']=(df['Start time'].duplicated(False)|df['End time'].duplicated(False)).astype(int)

Or did you need

df['isDup']=(df['Start time'].between(df['Start time'].shift(),df['End time'].shift())).astype(int)

Map the time like values in columns start_time and end_time to pandas TimeDelta objects and subtract 1 seconds from the 00:00:00 timedelta values in end_time column.

c = ['start_time', 'end_time']
s, e = df[c].astype(str).apply(pd.to_timedelta).to_numpy().T
e[e == pd.Timedelta(0)] += pd.Timedelta(days=1, seconds=-1)

Then for each pair of start_time and end_time in the dataframe df mark the corresponding duplicate intervals using numpy broadcasting:

m = (s[:, None] >= s) & (e[:, None] <= e)
np.fill_diagonal(m, False)
df['isDupe'] = (m.any(1) & ~df[c].duplicated(keep=False)).view('i1')

---

# example 1
  start_time  end_time  isDupe
0   00:12:38  01:00:02       0
1   00:55:14  01:00:02       1
2   01:00:02  01:32:40       0
3   01:00:02  01:08:40       1
4   01:41:22  03:56:23       0
5   18:58:26  19:16:49       0
6   20:12:37  20:52:49       0
7   20:55:16  22:02:50       0
8   22:21:24  22:48:50       0
9   23:11:30  00:00:00       0

# example 2
  start_time  end_time  isDupe
0   13:32:54  13:32:55       0
1   13:36:10  13:50:16       0
2   13:37:54  13:38:14       1
3   13:46:38  13:46:45       1
4   13:48:59  13:49:05       1
5   13:50:16  13:50:20       0
6   14:03:39  14:03:49       0
7   15:36:20  15:36:20       0
8   15:46:47  15:46:47       0