On Homotopy of connected graphs having the same degree function Original Research Article

Let Gα denote the set of simple graphs having the degree function α. It is known that any two graphs from Gα are homotopic in the sense that one can be obtained from the other by a series of simple two-edge switches so that every intermediate graph is in Gα. Let Ck denote the set of simple graphs having at most k components. A natural question arises whether it is possible to extend the above homotopy result on the class Ck∩Gα, and in particular, whether it is possible to obtain a connected simple graph from another connected simple graph of the same degree function by a series of two edge switches preserving connectivity. In this paper we give an answer to this question. We also consider two-edge switches of a special type and single out some classes of graphs for which these special operations provide the corresponding homotopy between its members.