The 
-path
complement graph 
is the graph complement of the path
graph 
.
The first few are illustrated above.
Since 
is self-complementary, 
is isomorphic to
.
Special cases are summarized in the table below.
 | graph name |
| 1 | singleton graph  |
| 2 | empty graph  |
| 3 |  |
| 4 | path graph  |
| 5 | house graph |
| 6 | tetragonal antiwedge graph |
has vertex count 
and edge count
 |
(1)
|
where 
is the binomial coefficient.
is connected for 
and Hamiltonian
for 
.
is planar for 
. Its exact graph
crossing numbers for 
, 8, 9, and 10 are 1, 3, 9, and 18, respectively. These values
suggest the conjecture
 |
(2)
|
for 
,
where
 |
(3)
|
and 
denotes the floor function. The values of 
for 
, 1, ... are 0, 0, 0, 0, 0, 1, 3, 9, 18, 36, 60, 100, ...
(OEIS A028723). The conjecture
agrees with the exact graph crossing numbers
for 
and matches computed upper bounds to at least 
(E. Weisstein, Oct. 1,
2026).
The simplex graph of the path complement graph 
is the Fibonacci
cube graph 
(Alikhani and Ghanbari 2024).
