Properties of Circuits in a W-Graph

Hua-An ZHAO  Wataru MAYEDA  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E77-A   No.10   pp.1692-1699
Publication Date: 1994/10/25
Online ISSN: 
Print ISSN: 0916-8508
Type of Manuscript: PAPER
Category: Graphs, Networks and Matroids
wild-component wi,  W-graph Ωw,  derived graph Gd,  inner path,  closed chain,  

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A W-graph is a partially known graph which contains wild-components. A wild-component is an incompletely defined connected subgraph having p vertices and p-1 unspecified edges. The informations we know on a wild-component are which has a vertex set and between any two vertices there is one and only one path. In this paper, we discuss the properties of circuits in a W-graph (called W-circuits). Although a W-graph has unspecified edges, we can obtain some important properties of W-circuits. We show that the W-ring sum of W-circuits is also a W-circuit in the same W-graph. The following (1) and (2) are proved: (1) A W-circuit Ci of a W-graph can be transformed into either a circuit or an edge disjoint union of circuits, denoted by Ci*, of a graph derived from the W-graph, (2) if W-circuits C1, C2, ・・・, Cn are linearly independent, then C1*, C2*, ・・・, Cn* obtained in (1) are also linearly independent.