Maps of Type K4 on Closed Two-Dimensional Manifolds

A map is a pair composed of a connected two-dimensional manifold and its subspace with the structure of a finite graph, and any connectivity component of the complement of this subspace is homeomorphic to an open two-dimensional disk. Two maps are called equivalent if there exists a homeomorphism of the manifold of the first map onto the manifold of the second map that isomorphically maps the graph of the first map onto the graph of the second map. One of the important problems of the map theory is to classify maps up to the accuracy of the introduced equivalence relation. This work deals with the classification of maps whose graphs are isomorphic to the graph K4. A set of eleven maps with their graphs isomorphic to the graph K4 is presented, and any map whose graph is isomorphic to the graph K4 is equivalent to some map of this set. Since the number of equivalence classes of maps with their graphs isomorphic to the graph K4 is known to be eleven, our set solves the classification problem (up to the accuracy of the introduced equivalence relation) for maps with their graphs isomorphic to the graph K4.