Graph properties checkable in linear time in the number of vertices

• Authors: Etienne Grandjean;Frédéric Olive
• Venue: Journal of Computer and System Sciences
• Year: 2004

This paper originates from the observation that many classical NP graph problems, including some NP-complete problems, are actually of very low nondeterministic time complexity. In order to formalize this observation, we define the complexity class vertex NLIN, which collects the graph problems computable on a nondeterministic RAM in time O(n). where n is the number of vertices of the input graph G = (V.E), rather than its usual size |V| + |E|. It appears that this class is robust (it is defined by a natural restrictive computational device: it is logically characterized by several simple fragments of existential second-order logic: it is closed under various combinatorial operators, including some restrictions of transitive closure) and meaningful (if contains natural NP problems: connectivity, hamiltonicity, non-planarity, etc.). Furthermore, the very restrictive definition of vertex NLIN seems to have beneficial effects on our ability to answer difficult questions about complexity lower bounds or separation between determinism and nondeterminism. For instance, we prove that vertex NLIN strictly contains its deterministic counterpart, vertex DLIN, and even that it does not coincide with its complementary class, co-vertex NLIN. Also, we prove that several famous graph problems (e.g. planarity, 2-colourability) do not belong to vertexNLIN, although they are computable in deterministic time O(|V| - |E|).