4 research outputs found
Decomposition theorem on matchable distributive lattices
A distributive lattice structure has been established on the
set of perfect matchings of a plane bipartite graph . We call a lattice {\em
matchable distributive lattice} (simply MDL) if it is isomorphic to such a
distributive lattice. It is natural to ask which lattices are MDLs. We show
that if a plane bipartite graph is elementary, then is
irreducible. Based on this result, a decomposition theorem on MDLs is obtained:
a finite distributive lattice is an MDL if and only if each factor
in any cartesian product decomposition of is an MDL. Two types of
MDLs are presented: and , where
denotes the cartesian product between -element
chain and -element chain, and is a poset implied by any
orientation of a tree.Comment: 19 pages, 7 figure
Block graphs of z-transformation graphs of perfect matchings of plane elementary bipartite graphs
Let G be a connected plane bipartite graph. The Z-transformation graph Z(G) is a graph where the vertices are the perfect matchings of G and where two perfect matchings are joined by an edge provided their symmetric difference is the boundary of an interior face of G. For a plane elementary bipartite graph G it is shown that the block graph of Z-transformation graph Z(G) is a path. As an immediate consequence, we have that Z(G) has at most two vertices of degree one