A number of special graph classes will be investigated. These classes include: line graphs of hypergraphs with prescribed properties; topological graphs; hereditary graph classes defined by prescribed relations between some invariants; locally defined classes.
The realization of the investigation should lead to the solution of some open problems in graph theory, to new methods of identification and analysis of promising classes of graph (hypergraph) and to the construction of new efficient algorithms.
It is supposed to investigate topological graphs with rectilinear embeddings of edges. An important question under consideration is to select plane subgraphs with given properties.
The following version of the well-known Trahtenbrot and Zykov problem (1963) will be considered. Given a graph-theoretic property P, is there a connected graph in which all vertex neighbourhoods induce a subgraph with the property P? For certain special properties P, the complete lists of P-local classes will be obtained.
Eulerian graphs form an important class of locally characterizable graphs. These graphs and their plane embeddings will be investigated in structural, enumerative and algorithmic aspects.
It is planned to investigate properties of graphs from the above classes, to construct efficient algorithms for solving some classical problems and to estimate the computational complexity of the membership recognition problem for these classes. Special attention will be paid to the existence of Hamiltonian circuits and constructing them efficiently.
The following main results are expected:
solution of the characterization and reconstructability problems for the class of line graphs of linear r-uniform hypergraphs (in particular, solution of a problem of L.Lovasz on characterizations of line graphs of 3-uniform hypergraphs) under some reasonable restrictions;
description of a number of topological graph classes admitting efficient algorithms for constructing subgraphs with prescribed properties;
characterizations of a number of perfect graph classes;
constructing new classes of strongly perfect graphs;
algorithms of recognizing the P4-structures of graphs with prescribed properties;
general methods of characterizing hereditary graph classes;
solution methods of the profile problem and other arrangement problems for some graph classes;
some methods of solving NP-hard problems for hereditary graph classes;
solution of a version of the Trahtenbrot and Zykov problem;
systematic description of enumeration methods for Eulerian graphs and maps;
locally sufficient conditions for Hamiltonicity.