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Let G be a plane bipartite graph and ${\cal M}(G)$ the set of perfect matchings of G. A property that the Z-transformation digraph of perfect matchings of G is acyclic implies a partially ordered relation on ${\cal M}(G)$ . It was shown that ${\cal M}(G)$ is a distributive lattice if G is (weakly) elementary. Based on the unit decomposition of alternating cycle systems, in this article...
An outerplane graph is a connected plane graph with all vertices lying on the boundary of its outer face. For a catacondensed benzenoid graph G, i.e. a 2-connected outerplane graph each inner face of which is a regular hexagon, S. Klavžar and P. Žigert [A min–max result on catacondensed benzenoid graphs, Appl. Math. Lett. 15 (2002) 279–283] discovered that the smallest number of elementary cuts that...
Let G be a plane bipartite graph and M(G) the set of perfect matchings of G. The Z-transformation graph of G is defined as a graph on M(G): M,M′∈M(G) are joined by an edge if and only if they differ only in one cycle that is the boundary of an inner face of G. A property that a certain orientation of the Z-transformation graph of G is acyclic implies a partially ordered relation on M(G). An equivalent...
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