On fixed edges and edge-reconstruction of series-parallel networks

A graph configuration is a vertex labeled simple graph. We say a configuration is exclusive for graphs with a property in a class of graphs if every graph in the class containing the configuration has the property. We use the method of exclusive configurations to search for graphs without fixed edges and forced edges investigate the fixed edges. An edge e of a graph G is said to be a fixed edge if G-e+e´≅G implies that e´=e, and a forced edge if G-e+e´ is an edge-reconstruction of G implies that e´=e. It is proved that all series-parallel networks contain fixed edges and forced edge except P₂×K₁ and P₃×K₁. A similar method is used to investigate the alternative wire problem in circuit design. Let G be a Boolean network and w a wire in G. Wire w´ is said to be an alternative wire of w if w´≠w and G-w+w´ has the same functionality as that of G. A wire is said to be fixed if it has no alternative wire. We present a set of exclusive configurations for Boolean networks in which every wire is fixed