On nonfeasible edge sets in matching-covered graphs

LetG=(V,E)be a matching-covered graph andXbe an edge set ofG.Xis said to be feasible if there exist two perfect matchingsM1andM2inGsuch that|M1 boolean AND X|not equivalent to|M2 boolean AND X| (mod 2). For anyV0 subset of V,Xis said to be switching-equivalent toX circle plus backward difference G(V0), where backward difference G(V0)is the set of edges inGeach of which has exactly one end inV0andA circle plus Bis the symmetric difference of two setsAandB. Lukot'ka and Rollova showed that whenGis regular and bipartite,Xis nonfeasible if and only ifXis switching-equivalent to null . This article extends Lukot'ka and Rollova's result by showing that this conclusion holds as long asGis matching-covered and bipartite. This article also studies matching-covered graphsGwhose nonfeasible edge sets are switching-equivalent to null orEand partially characterizes these matching-covered graphs in terms of their ear decompositions. Another aim of this article is to construct infinite manyr-connected andr-regular graphs of class 1 containing nonfeasible edge sets not switching-equivalent to either null orEfor an arbitrary integerrwithr >= 3, which provides a negative answer to a problem proposed by He et al.