ON THE REPRESENTATION OF THE RIGID SUB-SYSTEMS OF A PLANE LINK SYSTEM

A combinatorial characterization of the rigidity of plane link systems was fIrst established by Laman(9) (he used the term 'skeletal structure' instead of 'link syst,em' of this paper). This paper presents a combinatorial analysis method for the structures of plane link systems. More precisely, the proposed method affords a representa­ tion of the family of all the rigid sub-systems of a plane link system. A link system is a mechanical object composed of rods and joints(links). A rod is a rigid bar connected with some other rods by joints at its ends and it moves freely around a joint. A link system was first studied systematically by Laman(9) , who estab­ lished a graph-theoretic charact-erization of the rigidity of link systems on a 2-dimensional space. But the generalization of the results of Laman to the 3-dimensional case is not yet known. As for the further researches about this problem, we refer to (2)(4). Bolker-Crapo(3) presented a matroid-theoretic approach to the bracing problem on a one-story-building. Lovasz(12) reduced the pinning down problem of a plane link system to a matroid parity problem. This paper presents a method for the representation of the inner struc­ tures of a plane link system. The main tool used in this paper is based on matroid theory.