On PM-compact Characterizations And Removable Edges In Matching Covered Graphs

A connected graph G with at least two vertices is matching covered,if any edge of G is contained in a perfect matching of G.The matching lattice of graph G is the set of all integer linear combinations of all incidence vectors of perfect matching of G.A matching covered graph G is extremal if the number of perfect matchings of G amounts to the dimension of the matching lattice of G.A graph is PM-compact,if the remaining graph has no perfect matching after removing two vertices disjoint alternative cycles arbitrarily.An edge e of a matching covered G is removable if G-e is still matching covered.A Halin graph is a graph G:=T U C,where T is a plane tree with at least four vertices in which no vertex has degree two,and C is a cycle connecting the pendant vertices of T in the cyclic order determined by the embedding of T.In this paper,we first prove that all extremal bricks are PM-compact.Moreover we obtain a lower bound on the number of removable edges of Halin graphs with n vertices is 1/4(n+2).