The completeness of the modal logic S4 for all topological spaces as well as for the real line R, the n-dimensional Euclidean space R and the segment (0, 1) etc. (with 2 interpreted as interior) was proved by McKinsey and Tarski in 1944. Several simplified proofs contain gaps. A new proof presented here combines the ideas published later by G. Mints and M. Aiello, J. van Benthem, G. Bezhanishvi...

A set of vertices S in a graph is convex if it contains all vertices which belong to shortest paths between vertices in S. The convexity number c(G) of a graph G is the maximum cardinality of a convex set of vertices which does not contain all vertices of G. We prove NP-completeness of the problem to decide for a given bipartite graph G and an integer k whether c(G) ≥ k. Furthermore, we identif...

An s-graph is a graph with two kinds of edges: subdivisible edges and real edges. A realisation of an s-graph B is any graph obtained by subdividing subdivisible edges of B into paths of arbitrary length (at least one). Given an s-graph B, we study the decision problem ΠB whose instance is a graph G and question is “Does G contain a realisation of B as an induced subgraph?”. For several B’s, th...

We consider two graph invariants that are used as a measure of nonplanarity: the splitting number of a graph and the size of a maximum planar subgraph. The splitting number of a graph G is the smallest integer k 0, such that a planar graph can be obtained from G by k splitting operations. Such operation replaces a vertex v by two nonadjacent vertices v1 and v2, and attaches the neighbors of v e...

The main mathematical result of this work is a quite simple formulation and proof of a Rasiowa-Sikorski-like theorem for countable lattices. Then the paper suggests an interpretation of this mathematical result as a completeness theorem for the formal topologies introduced by G. Sambin in order to provide a constructive approach to topology which is expressible within Martin Löf’s intuitionisti...