By W. D. Wallis
Concisely written, light creation to graph idea appropriate as a textbook or for self-study Graph-theoretic functions from various fields (computer technology, engineering, chemistry, administration technology) second ed. comprises new chapters on labeling and communications networks and small worlds, in addition to multiplied beginner's fabric Many extra adjustments, advancements, and corrections caused by lecture room use
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Additional info for A Beginner's Guide to Graph Theory
There are three bridges leading to the area C: you can traverse two of these, one leading into C and the other leading out, at one time in your tour. There is only one bridge left: if you cross it going into C, then you cannot leave C again, unless you use one of the bridges twice, so C must be the finish of the walk; if you cross it in the other direction, C must have been the start of the walk. In either event, C is either the place where you started or the place where you finished. A similar analysis can be applied to A, Band D, since each has an odd number of bridges.
32 2. Walks, Paths and Cycles A C D F 1 00- o -D L Fig. 11. Construct an Euler walk We start with ACBE A. We repla ce C by CEDGEI FC , yielding AC EDGEl FCBEA, then replace the first E by E GIL IE, with the result ACEDGH l LH IGEl FCBEA. Finally I is replaced by I K L 1, and the Euler walk is ACEDGH l LH I KLIGEl FCBEA . , C G ~' K L J (b) G J oC ' 0 Ko of L (c) Fig. 12. Constructing an Euler walk A good application of Euler walks is planning the route of a highway inspector or mail contractor, who must travel over all the roads in a highway system.
Then unite the two walks as follows: at one place where the original walk contained c, insert the new walk. For example, if the two walks are x,y, ... ,z,c,u, ... ,x and c, s, ... , t, C, then the resulting walk will be x, y, ... , z, c, s, ... , t, c, u, ... , x. (There may be more than one possible answer, if c occurred more than once in the first walk. ) The new walk is a closed simple walk in the original multigraph. Repeat the process of deletion, this time deleting the newly formed walk. Continue in this way.
A Beginner's Guide to Graph Theory by W. D. Wallis