Modal Logics for Finite Graphs

Abstract

We present modal logics for four classes of finite graphs: finite directed graphs, finite acyclic directed graphs, finite undirected graphs and finite loopless undirected graphs. For all these modal proof theories we discuss soundness and completeness results with respect to each of these classes of graphs. Moreover, we investigate whether some well-known properties of undirected graphs are modally definable or not: k-colouring, planarity, connectivity and properties that a graph is Eulerian or Hamiltonian. Finally, we present an axiomatization for colouring and prove that it is sound and complete with respect to the class of finite k-colourable graphs. One of most interesting feature of this approach is the use of the axioms of Dynamic Logic together with the Lob axiom to ensure acyclicity.