# Is finding the longest path of a graph NP-complete?

The problem of finding the largest subgraph of a graph that has a Hamiltonian path can be restated as finding the longest path of a graph. Is this NP-complete? Also, is finding the $k$-length path of a graph NP-complete? Is it still NP-complete if we require the path to visit a given vertex?

• is the longest path as you quoted required to be simple or not? – Charlie Parker Apr 29 '16 at 2:40

First, it is easy to see that the problem is in $\text{NP}$. The longest path is a Hamiltonian one since it visits all vertices. Indeed, there is a straightforward reduction from $\text{HAM-PATH}$ to it. For details and some special cases, see for example here. Likewise, it is $\text{NP}$-complete to decide whether there is a path of length $k$ with the a similar argument: just set $k$ such that the problem is equivalent of solving $\text{HAM-PATH}$. Finally, if I understood your third point correctly, it doesn't make the problem any easier if you in addition require the path to visit a specific vertex $v$. Similar reasoning holds for this case.