Applications of graph theory

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I have two major projects.  One is to find as quickly as possible the minimum cycle basis of a graph.  Cycle bases are used to study cycles in networks, especially when the networks lead to sets of equations.  Smaller bases produce simpler equations to be solved.  Applications include electrical networks, structural engineering, scheduling subway systems, studying metabolic pathways, in topology.       The other project is the study of clause trees.  These give a way to represent a restricted set of logical proofs in simple diagrams, that can be represented in a computer.  Then computer programs can be written, starting from known axioms, to construct and manipulate clause trees to find proofs of theorems. Few new results have been found using such automated theorem provers yet, but such systems are used in many artifical intelligence applications.  Can clause trees be used to find proofs faster than other methods of reasoning with computers?