Complexity of compatible decompositions of eulerian graphs and their transformations

Abstract

In this paper some earlier defined local transformations between eulerian trails are generalized to transformations between decompositions of graphs into (possibly more) closed subtrails. For any graph G with a forbidden partition system F, we give an efficient algorithm which transforms any F-compatible decomposition of G into closed subtrails to another one, and at the same time it preserves F-compatibility and does not increase the number of subtrails by more than one. From this, several earlier results for eulerian trails easily follow. These results are embedded into the rich spectrum of results of theory of eulerian graphs and their applications. We further apply this statement to digraphs and discuss the time complexity of enumeration of all F-compatible decompositions (resp. of all F-compatible eulerian trails) in both graphs and digraphs.