A Logical Approach to Hamiltonian Graphs

Abstract

Graphs are among the most frequently used structures in computer science. A lot of problems can be modelled using a graph and can then be solved by checking whether the graph satisfies some property. In this work, we are interested in how to use logical frameworks as a generic tool to express and efficiently check graph properties. In order to reason about this, we choose to analyze the Hamiltonian property and choose the family of modal logics as our framework. Our analysis has to deal with two central issues: whether each of the modal languages under consideration has enough expressive power to describe this property and how complex (computationally) it is to use these logics to actually test whether a given graph has this property. First, we show that this property is not definable in a basic modal logic or in any bisimulation-invariant extension of it, like the modal μ-calculus. We then show that it is possible to express it in a basic hybrid logic. Unfortunately, the Hamiltonian property still cannot be efficiently checked in this logic. In a second attempt, we extend this basic hybrid logic with the ↓ operator and show that we can check the Hamiltonian property with optimal (NP-Complete) complexity in this logic.