A weighted pair graph representation for reconstructibility of Boolean control networks

A new concept of weighted pair graphs (WPGs) is proposed to represent a new reconstructibility definition for Boolean control networks (BCNs), which is a generalization of the reconstructibility definition given in [Fornasini & Valcher, TAC2013, Def. 4]. Based on the WPG representation, an effective algorithm for determining the new reconstructibility notion for BCNs is designed with the help of the theories of finite automata and formal languages. We prove that a BCN is not reconstructible iff its WPG has a complete subgraph. Besides, we prove that a BCN is reconstructible in the sense of [Fornasini & Valcher, TAC2013, Def. 4] iff its WPG has no cycles, which is simpler to be checked than the condition in [Fornasini & Valcher, TAC2013, Thm. 4].